#### Chapters

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solution of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

## Chapter 17: Increasing and Decreasing Functions

#### Chapter 17: Increasing and Decreasing Functions solutions [Page 10]

Prove that the function f(x) = log_{e} x is increasing on (0, ∞) ?

Prove that the function f(x) = log_{a} x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?

Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?

Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?

Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?

Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?

Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?

Without using the derivative, show that the function *f* (*x*) = | *x* | is.

(a) strictly increasing in (0, ∞)

(b) strictly decreasing in (−∞, 0) .

Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?

#### Chapter 17: Increasing and Decreasing Functions solutions [Pages 33 - 35]

Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x^{2 }?

Find the interval in which the following function are increasing or decreasing f(x) = x^{2} + 2x − 5 ?

Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x^{2 }?

Find the interval in which the following function are increasing or decreasing f(x) = 2x^{3} − 12x^{2} + 18x + 15 ?

Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x^{2} − 2x^{3 }?

Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x^{2} − 2x^{3 }?

Find the interval in which the following function are increasing or decreasing *f*(*x*) = 5*x*^{3} − 15*x*^{2} − 120*x* + 3 ?

Find the interval in which the following function are increasing or decreasing f(x) = x^{3} − 6x^{2} − 36x + 2 ?

Find the interval in which the following function are increasing or decreasing f(x) = 2x^{3} − 15x^{2} + 36x + 1 ?

Find the interval in which the following function are increasing or decreasing f(x) = 2x^{3} + 9x^{2} + 12x + 20 ?

Find the interval in which the following function are increasing or decreasing f(x) = 2x^{3} − 9x^{2} + 12x − 5 ?

Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x^{2} − 2x^{3} ?

Find the interval in which the following function are increasing or decreasing f(x) = 2x^{3} − 24x + 107 ?

Find the interval in which the following function are increasing or decreasing f(x) = −2x^{3} − 9x^{2} − 12x + 1 ?

Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)^{2 }?

Find the interval in which the following function are increasing or decreasing f(x) = x^{3} − 12x^{2} + 36x + 17^{ }?

Find the interval in which the following function are increasing or decreasing f(x) = 2x^{3} − 24x + 7 ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?

Find the interval in which the following function are increasing or decreasing f(x) = x^{4} − 4x ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?

Find the interval in which the following function are increasing or decreasing f(x) = x^{4} − 4x^{3} + 4x^{2} + 15 ?

Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] *x* > 0 ?

Find the interval in which the following function are increasing or decreasing f(x) = x^{8} + 6x^{2 }?

Find the interval in which the following function are increasing or decreasing f(x) = x^{3} − 6x^{2} + 9x + 15 ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?

Determine the values of *x* for which the function f(x) = x^{2} − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x^{2} − 6x + 9 where the normal is parallel to the line y = x + 5 ?

Find the intervals in which *f*(*x*) = sin *x* − cos *x*, where 0 < *x* < 2π is increasing or decreasing ?

Show that f(x) = e^{2x} is increasing on R ?

Show that f(x) = e^{1}^{/x}, x ≠ 0 is a decreasing function for all x ≠ 0 ?

Show that f(x) = log_{a} x, 0 < a < 1 is a decreasing function for all x > 0 ?

Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?

Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?

Show that f(x) = x − sin x is increasing for all x ∈ R ?

Show that f(x) = x^{3} − 15x^{2} + 75x − 50 is an increasing function for all x ∈ R ?

Show that f(x) = cos^{2} x is a decreasing function on (0, π/2) ?

Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?

Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π) ?

Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?

Show that f(x) = tan^{−1} (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?

Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?

Show that the function f(x) = cot \[-\] ^{l}(sin*x* + cos*x*) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?

Show that f(x) = (x − 1) e^{x} + 1 is an increasing function for all x > 0 ?

Show that the function x^{2} − x + 1 is neither increasing nor decreasing on (0, 1) ?

Show that f(x) = x^{9} + 4x^{7} + 11 is an increasing function for all x ∈ R ?

Prove that the function f(x) = x^{3} − 6x^{2} + 12x − 18 is increasing on R ?

State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x^{2} − 6x + 3 is increasing on the interval [4, 6] ?

Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4) ?

Show that f(x) = tan^{−1} x − x is a decreasing function on R ?

Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?

Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?

Find the intervals in which f(x) = (x + 2) e^{−x} is increasing or decreasing ?

Show that the function f given by f(x) = 10^{x} is increasing for all x ?

Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?

Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?

Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?

Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?

Prove that the function f given by f(x) = x^{3} − 3x^{2} + 4x is strictly increasing on R ?

Prove that the function f(x) = cos x is:

(i) strictly decreasing in (0, π)

(ii) strictly increasing in (π, 2π)

(iii) neither increasing nor decreasing in (0, 2π).

Show that f(x) = x^{2} − x sin x is an increasing function on (0, π/2) ?

Find the value(s) of a for which f(x) = x^{3} − ax is an increasing function on R ?

Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?

Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?

Let f defined on [0, 1] be twice differentiable such that | f"(x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?

Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?

Find the interval in which *f*(*x*) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?

Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?

#### Chapter 17: Increasing and Decreasing Functions solutions [Pages 39 - 40]

What are the values of 'a' for which f(x) = a^{x} is increasing on R ?

What are the values of 'a' for which f(x) = a^{x} is decreasing on R ?

Write the set of values of 'a' for which f(x) = log_{a} x is increasing in its domain ?

Write the set of values of 'a' for which f(x) = log_{a} x is decreasing in its domain ?

Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?

Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?

Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?

Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?

Write the set of values of *k* for which f(x) = kx − sin x is increasing on R ?

If g (x) is a decreasing function on R and f(x) = tan^{−1} [g (x)]. State whether f(x) is increasing or decreasing on R ?

Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?

Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?

State whether f(x) = tan x − x is increasing or decreasing its domain ?

Write the set of values of *a* for which f(x) = cos x + a^{2} x + b is strictly increasing on R ?

#### Chapter 17: Increasing and Decreasing Functions solutions [Pages 40 - 42]

The interval of increase of the function *f*(*x*) = *x* − *e ^{x}* + tan (2π/7) is

(a) (0, ∞)

(b) (−∞, 0)

(c) (1, ∞)

(d) (−∞, 1)

The function *f*(*x*) = cot^{−1} *x* + *x* increases in the interval

(a) (1, ∞)

(b) (−1, ∞)

(c) (−∞, ∞)

(d) (0, ∞)

The function *f*(*x*) = *x ^{x}* decreases on the interval

(a) (0,

*e*)

(b) (0, 1)

(c) (0, 1/

*e*)

(d) none of these

The function *f*(*x*) = 2 log (*x* − 2) − *x*^{2} + 4*x* + 1 increases on the interval

(a) (1, 2)

(b) (2, 3)

(c) (1, 3)

(d) (2, 4)

If the function *f*(*x*) = 2*x*^{2} − *kx* + 5 is increasing on [1, 2], then *k* lies in the interval

(a) (−∞, 4)

(b) (4, ∞)

(c) (−∞, 8)

(d) (8, ∞)

Let f(x) = x^{3}^{ }+ ax^{2} + bx + 5 sin^{2}x be an increasing function on the set R. Then, a and bsatisfy

(a) a^{2} − 3b − 15 > 0

(b) a^{2} − 3b + 15 > 0

(c) a^{2} − 3b + 15 < 0

(d) a > 0 and b > 0

The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:

(a) even and increasing

(b) odd and increasing

(c) even and decreasing

(d) odd and decreasing

If the function f(x) = 2 tan x + (2a + 1) log_{e} | sec x | + (a − 2) x is increasing on R, then

(a) a ∈ (1/2, ∞)

(b) a ∈ (−1/2, 1/2)

(c) a = 1/2

(d) a ∈ R

Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is

(a) increasing on (0, π/2)

(b) decreasing on (0, π/2)

(c) increasing on (0, π/4) and decreasing on (π/4, π/2)

(d) none of these

Let f(x) = x^{3} − 6x^{2} + 15x + 3. Then,

(a) f(x) > 0 for all x ∈ R

(b) f(x) > f(x + 1) for all x ∈ R

(c) f(x) is invertible

(d) none of these

The function f(x) = x^{2} e^{−x} is monotonic increasing when

(a) x ∈ R − [0, 2]

(b) 0 < x < 2

(c) 2 < x < ∞

(d) x < 0

Function *f*(*x*) = cos *x* − 2 λ *x* is monotonic decreasing when

(a) λ > 1/2

(b) λ < 1/2

(c) λ < 2

(d) λ > 2

In the interval (1, 2), function *f*(*x*) = 2 | *x* − 1 | + 3 | *x* − 2 | is

(a) monotonically increasing

(b) monotonically decreasing

(c) not monotonic

(d) constant

Function *f*(*x*) = *x*^{3} − 27*x* + 5 is monotonically increasing when

(a) *x* < −3

(b) | *x* | > 3

(c) *x* ≤ −3

(d) | *x* | ≥ 3

Function f(x) = 2x^{3} − 9x^{2} + 12x + 29 is monotonically decreasing when

(a) x < 2

(b) x > 2

(c) x > 3

(d) 1 < x < 2

If the function f(x) = kx^{3} − 9x^{2} + 9x + 3 is monotonically increasing in every interval, then

(a) k < 3

(b) k ≤ 3

(c) k > 3

(d) k ≥ 3

f(x) = 2x − tan^{−1} x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when

(a) *x* > 0

(b) *x* < 0

(c) *x* ∈ *R*

(d) *x* ∈ *R* − {0}

Function f(x) = | x | − | x − 1 | is monotonically increasing when

(a) x < 0

(b) x > 1

(c) x < 1

(d) 0 < x < 1

Every invertible function is

(a) monotonic function

(b) constant function

(c) identity function

(d) not necessarily monotonic function

In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is

(a) increasing

(b) decreasing

(c) constant

(d) none of these

If the function *f*(*x*) = cos |*x*| − 2*ax* + *b* increases along the entire number scale, then

(a) *a* = *b*

(b) \[a = \frac{1}{2}b\]

(c) \[a \leq - \frac{1}{2}\]

(d) \[a > - \frac{3}{2}\]

The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is

(a) strictly increasing

(b) strictly decreasing

(c) neither increasing nor decreasing

(d) none of these

The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if

(a) λ < 1

(b) λ > 1

(c) λ < 2

(d) λ > 2

Function f(x) = a^{x} is increasing on R, if

(a) a > 0

(b) a < 0

(c) 0 < a < 1

(d) a > 1

Function f(x) = log_{a} x is increasing on R, if

(a) 0 < a < 1

(b) a > 1

(c) a < 1

(d) a > 0

Let Ï•(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, Ï• (x)

(a) increases on [0, a]

(b) decreases on [0, a]

(c) increases on [−a, 0]

(d) decreases on [a, 2a]

If the function f(x) = x^{2} − kx + 5 is increasing on [2, 4], then

(a) k ∈ (2, ∞)

(b) k ∈ (−∞, 2)

(c) k ∈ (4, ∞)

(d) k ∈ (−∞, 4).

The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is

(a) increasing

(b) decreasing

(c) constant

(d) none of these

If the function f(x) = x^{3} − 9kx^{2} + 27x + 30 is increasing on R, then

(a) −1 ≤ k < 1

(b) k < −1 or k > 1

(c) 0 < k < 1

(d) −1 < k < 0

The function f(x) = x^{9} + 3x^{7} + 64 is increasing on

(a) R

(b) (−∞, 0)

(c) (0, ∞)

(d) R_{0}

#### Chapter 17: Increasing and Decreasing Functions solutions [Page 7]

f ( \[x\])=4\[x^2\]-4 \[x\] + 4 on R .

f(x)=(x-1)^{2}+2 on R ?

f(x)=| x+2 | on R .

f(x)=sin 2x+5 on R .

f(x) = | sin 4x+3 | on R ?

f(x)=2x^{3}^{ }+5 on R .

f (x) = \[-\] | x + 1 | + 3 on R .

f(x) = 16x^{2} \[-\] 16x + 28 on R ?

f(x) = x^{3 }\[-\] 1 on R .

#### Chapter 17: Increasing and Decreasing Functions solutions [Page 16]

f(x) = (x \[-\] 5)^{4}.

f(x) = x^{3 }\[-\] 3x .

f(x) = x^{3} (x \[-\] 1)^{2 }.

f(x) = (x \[-\] 1) (x+2)^{2}.

f(x) = \[\frac{1}{x^2 + 2}\] .

f(x) = x^{3 }\[-\] 6x^{2} + 9x + 15 .

f(x) = sin 2x, 0<x< \[\pi\] .

f(x) = sin x \[-\] cos x, 0 < x<2 \[\pi\] .

f(x) = cos x, 0<x< \[\pi\] .

f(x) = sin 2x \[-\] x, \[- \frac{\pi}{2} < \frac{<}{}x\frac{<}{}\frac{\pi}{2}\] .

f(x) = 2sin x\[-\] x, \[- \frac{\pi}{2} < \frac{<}{}x\frac{<}{}\frac{\pi}{2}\] .

f(x) =\[x\sqrt{1 - x} , x > 0\].

f(x) = x^{3} (2x \[-\] 1)^{3}.

f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .

#### Chapter 17: Increasing and Decreasing Functions solutions [Page 31]

f(x) = x^{4} \[-\] 62x^{2} + 120x + 9.

f(x) = x^{3}\[-\] 6x^{2} + 9x + 15

f(x) = (x \[-\] 1) (x+2)^{2}.

f(x) = 2/x \[-\] 2/x^{2} , x>0 .

f(x) = xe^{x}.

f(x) = x/2+2/x, x>0 .

f(x) = (x+1) (x+2)^{1/3}, \[x\frac{>}{} - 2\] .

f(x) = \[x\sqrt{32 - x^2}, - 5\frac{<}{}x\frac{<}{}5\] .

f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .

f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .

f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .

f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .

f(x) = (x \[-\] 1) (x \[-\] 2)^{2}.

f(x) = \[x\sqrt{1 - x} , x\frac{<}{}1\] .

f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .

The function y = a log x+bx^{2} + x has extreme values at x=1 and x=2. Find a and b ?

Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?

Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]

Find the maximum and minimum values of y = tan \[x - 2x\] .

If f(x) = x^{3} + ax^{2} + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?

Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?

#### Chapter 17: Increasing and Decreasing Functions solutions [Page 37]

f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .

f(x) = (x \[-\] 1)^{2} + 3 in [ \[-\] 3,1] ?

f(x) = 3x4 \[-\] 8x^{3} + 12x^{2}\[-\] 48x + 25 in [0,3] .

f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .

Find the maximum value of 2x^{3}\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].

Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .

Find the absolute maximum and minimum values of a function f given by \[f(x) = 12 x^{4/3} - 6 x^{1/3} , x \in [ - 1, 1]\] .

Find the absolute maximum and minimum values of a function f given by \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?

#### Chapter 17: Increasing and Decreasing Functions solutions [Pages 72 - 74]

Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.

Divide 64 into two parts such that the sum of the cubes of two parts is minimum.

How should we choose two numbers, each greater than or equal to \[-\] 2, whose sum______________ so that the sum of the first and the cube of the second is minimum?

Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.

Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm^{3}, which has the minimum surface area?

A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .

Find the point at which M is maximum in case.

A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .

Find the point at which M is maximum in case.

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?

A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?

Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.

Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.

Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\] What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.

A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?

A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is 2 m and volume is 8 m^{3}. If building of tank cost 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.

A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.

Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]

A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?

Prove that a conical tent of given capacity will require the least amount of canavas when the height is \[\sqrt{2}\] times the radius of the base.

Show that the cone of the greatest volume which can be inscribed in a given spher has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .

An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius *a*. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .

Prove that the least perimeter of an isosceles triangle in which a circle of radius *r* can be inscribed is \[6\sqrt{3}\]r.

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?

Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?

A closed cylinder has volume 2156 cm^{3}. What will be the radius of its base so that its total surface area is minimum ?

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]

Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .

Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?

Find the point on the curve y2=4x which is nearest to the point (2,\[-\] 8).

Find the point on the curve x^{2}=8y which is nearest to the point (2,4) ?

Find the point on the parabolas x^{2} = 2y which is closest to the point (0,5) ?

Find the coordinates of a point on the parabola y=x^{2}+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?

Find the point on the curvey y^{2}=2x which is at a minimum distance from the point (1,4).

Find the maximum slope of the curve y= \[- x^3 + 3 x^2 + 2x - 27 .\]

The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] ind the daily output to maximum the total profit.

Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.

An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.

A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?

The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\] .

The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?

A straight line is drawn through a given point P(1,4). Determine the least value of the sum of the intercepts on the coordinate axes ?

The total area of a page is 150 cm^{2}. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?

The space s described in time *t *by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.

A particle is moving in a straight line such that its distance at any time *t* is given by S = \[\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .\] Find when its velocity is maximum and acceleration minimum.

#### Chapter 17: Increasing and Decreasing Functions solutions [Page 80]

Write necessary condition for a point x = c to be an extreme point of the function f(x).

Write sufficient conditions for a point x=c to be a point of local maximum.

If f(x) attains a local minimum at x=c, then write the values off' (c) and f'' (c).

Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]

Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]

Write the point where f(x) = x log, x attains minimum value.

Find the least value of f(x) =\[ax + \frac{b}{x}\], where a>0, b>0 and x>0 .

Write the minimum value of f(x) = x^{x} .

Write the maximum value of f(x) = x^{1}^{/x}.

Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .

#### Chapter 17: Increasing and Decreasing Functions solutions [Pages 80 - 82]

The maximum value of x^{1}^{/x}, x>0 is

(a) e^{1}^{/e}

^{(b) \[\left( \frac{1}{e} \right)^e\]}

(c) 1

(d) none of these

If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then

(a) \[ab < \frac{c^2}{4}\]

(b) \[ab \frac{>}{} \frac{c^2}{4}\]

(c) \[ab \frac{>}{} \frac{c^{}}{4}\]

The minimum value of \[\frac{x}{\log_e x}\] is

(a) e

(b) 1/e

(c) 1

(d) none of these

For the function f(x) = \[x + \frac{1}{x}\]

(a) x = 1 is a point of maximum

(b) x = \[-\] 1 is a point of minimum

(c) maximum value > minimum value

(d) maximum value< minimum value

Let f(x) = x^{3}+3x^{2 }\[-\] 9x+2. Then, f(x) has

(a) a maximum at x = 1

(b) a minimum at x = 1

(c) neither a maximum nor a minimum at x = \[-\] 3

(d) none of these

The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is

(a) 6

(b) 4

(c) 8

(d) none of these

The number which exceeds its square by the greatest possible quantity is

(a) \[\frac{1}{2}\]

(b) \[\frac{1}{4}\]

(c) \[\frac{3}{4}\]

(d) none of these

Let f(x) = (x \[-\] a)^{2} + (x \[-\] b)^{2} + (x \[-\] c)^{2}. Then, f(x) has a minimum at x =

(a) \[\frac{a + b + c}{3}\]

(b) \[\sqrt[3]{abc}\]

(c) \[\frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}\]

(d) none of these

The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is

(a) \[\frac{1}{4}\]

(b) \[\frac{1}{2}\]

(c) \[\frac{1}{8}\]

(d) none of these

The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)^{2} assumes minimum value at x =

(a) 5

(b) \[\frac{5}{2}\]

(c) 3

(d) 2

At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is

(a) 0

(b) maximum

(c) minimum

(d) none of these

If x lies in the interval [0,1], then the least value of x2 + x + 1 is

(a) 3

(b) \[\frac{3}{4}\]

(c) 1

(d) none of these

The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is

(a) 126

(b) 135

(c) 160

(d) 0

The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is

(a) \[- \frac{1}{4}\]

(b) \[- \frac{1}{3}\]

(c) \[\frac{1}{6}\]

(d) \[\frac{1}{5}\]

The point on the curve y^{2} = 4x which is nearest to, the point (2,1) is

(a) \[1, 2\sqrt{2}\]

(b) (1,2)

(c) (1,\[-\] 2)

(d) ( \[-\] 2,1)

If x+y=8, then the maximum value of xy is

(a) 8

(b) 16

(c) 20

(d) 24

The least and greatest values of f(x) = x^{3}\[-\] 6x^{2}+9x in [0,6], are

(a) 3,4

(b) 0,6

(c) 0,3

(d) 3,6

f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x =

(a) \[\frac{\pi}{3}\]

(b) \[\frac{\pi}{4}\]

(c) \[\frac{\pi}{6}\]

(d) 0

If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is

(a) \[\frac{3}{4}\]

(b) \[\frac{1}{3}\]

(c) \[\frac{1}{4}\]

(d) \[\frac{2}{3}\]

The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is

(a) 75

(b) 50

(c) 25

(d) 55

If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is

(a) \[-\] 2

(b) 0

(c) 3

(d) none of these

If(x) = \[\frac{1}{4x2 + 2x + 1}\] then its maximum value is

(a) \[\frac{4}{3}\]

(b) \[\frac{2}{3}\]

(c) 1

(d) \[\frac{3}{4}\]

Let x, y be two variables and x>0, xy=1, then minimum value of x+y is

(a) 1

(b) 2

(c) \[2\frac{1}{2}\]

(d) \[3\frac{1}{3}\]

f(x) = 1+2 sin x+3 cos^{2}x, \[0\frac{<}{}x\frac{<}{}\frac{2\pi}{3}\] is

(a) Minimum at x =\[\frac{\pi}{2}\]

(b) Maximum at x = sin \[- 1\] ( \[\frac{1}{\sqrt{3}}\])

(c) Minimum at x = \[\frac{\pi}{6}\]

(d) Maximum at sin** **\[- 1\] (\[\frac{1}{6})\]

The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x =

(a) 3

(b) 0

(c) 4

(d) 2

The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is

(a) \[- \frac{1}{4}\]

(b) \[- \frac{1}{3}\]

(c) \[\frac{1}{6}\]

(d) \[\frac{1}{5}\]

Let f(x) = 2x^{3}\[-\] 3x^{2}\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x =

(a) \[-\] 2

(b) \[-\]1

(c) 2

(d) 4

The minimum value of x log_{e} x is equal to

(a) e

(b) 1/e

(c) \[-\] 1/e

(d) 2/e

(e) \[-\] e

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

#### Textbook solutions for Class 12

## RD Sharma solutions for Class 12 Mathematics chapter 17 - Increasing and Decreasing Functions

RD Sharma solutions for Class 12 Maths chapter 17 (Increasing and Decreasing Functions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Mathematics chapter 17 Increasing and Decreasing Functions are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

Using RD Sharma Class 12 solutions Increasing and Decreasing Functions exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 17 Increasing and Decreasing Functions Class 12 extra questions for Maths and can use shaalaa.com to keep it handy for your exam preparation