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

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

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

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

Chapter 17: Increasing and Decreasing Functions

Ex. 17.1Ex. 17.2Others

Chapter 17: Increasing and Decreasing Functions Exercise 17.1 solutions [Page 10]

Ex. 17.1 | Q 1 | Page 10

Prove that the function f(x) = loge x is increasing on (0, ∞) ?

Ex. 17.1 | Q 2 | Page 10

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

Ex. 17.1 | Q 3 | Page 10

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

Ex. 17.1 | Q 4 | Page 10

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

Ex. 17.1 | Q 5 | Page 10

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

Ex. 17.1 | Q 6 | Page 10

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

Ex. 17.1 | Q 7 | Page 10

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

Ex. 17.1 | Q 8 | Page 10

Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .

Ex. 17.1 | Q 9 | Page 10

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

Chapter 17: Increasing and Decreasing Functions Exercise 17.2 solutions [Pages 33 - 35]

Ex. 17.2 | Q 1.01 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2  ?

Ex. 17.2 | Q 1.02 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = x2 + 2x − 5  ?

Ex. 17.2 | Q 1.03 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = 6 − 9x − x2  ?

Ex. 17.2 | Q 1.04 | Page 33

Find the interval in which the following function are increasing or decreasing   f(x) = 2x3 − 12x2 + 18x + 15 ?

Ex. 17.2 | Q 1.05 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x?

Ex. 17.2 | Q 1.06 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x?

Ex. 17.2 | Q 1.07 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = 5x3 − 15x2 − 120x + 3 ?

Ex. 17.2 | Q 1.08 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?

Ex. 17.2 | Q 1.09 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?

Ex. 17.2 | Q 1.1 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20  ?

Ex. 17.2 | Q 1.11 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?

Ex. 17.2 | Q 1.12 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?

Ex. 17.2 | Q 1.13 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = 2x3 − 24x + 107  ?

Ex. 17.2 | Q 1.14 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1  ?

Ex. 17.2 | Q 1.15 | Page 33

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

Ex. 17.2 | Q 1.16 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?

Ex. 17.2 | Q 1.17 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = 2x3 − 24x + 7 ?

Ex. 17.2 | Q 1.18 | Page 33

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

Ex. 17.2 | Q 1.19 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?

Ex. 17.2 | Q 1.2 | Page 33

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

Ex. 17.2 | Q 1.21 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = x4 − 4x3 + 4x2 + 15 ?

Ex. 17.2 | Q 1.22 | Page 33

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 ?

Ex. 17.2 | Q 1.23 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2  ?

Ex. 17.2 | Q 1.24 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?

Ex. 17.2 | Q 1.25 | Page 33

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

Ex. 17.2 | Q 1.26 | Page 33

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

Ex. 17.2 | Q 1.27 | Page 33

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

Ex. 17.2 | Q 1.28 | Page 33

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

Ex. 17.2 | Q 2 | Page 34

Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ? 

Ex. 17.2 | Q 3 | Page 34

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

Ex. 17.2 | Q 4 | Page 34

Show that f(x) = e2x is increasing on R ?

Ex. 17.2 | Q 5 | Page 34

Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?

Ex. 17.2 | Q 6 | Page 34

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

Ex. 17.2 | Q 7 | Page 34

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) ?

Ex. 17.2 | Q 8 | Page 34

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

Ex. 17.2 | Q 9 | Page 34

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

Ex. 17.2 | Q 10 | Page 34

Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?

Ex. 17.2 | Q 11 | Page 34

Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?

Ex. 17.2 | Q 12 | Page 34

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

Ex. 17.2 | Q 13 | Page 34

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).

Ex. 17.2 | Q 14 | Page 34

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

Ex. 17.2 | Q 15 | Page 34

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

Ex. 17.2 | Q 16 | Page 34

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

Ex. 17.2 | Q 17 | Page 34

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

Ex. 17.2 | Q 18 | Page 34

Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?

Ex. 17.2 | Q 19 | Page 34

Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?

Ex. 17.2 | Q 20 | Page 34

Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ? 

Ex. 17.2 | Q 21 | Page 35

Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?

Ex. 17.2 | Q 22 | Page 35

State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?

Ex. 17.2 | Q 23 | Page 35

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

Ex. 17.2 | Q 24 | Page 35

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

Ex. 17.2 | Q 25 | Page 35

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

Ex. 17.2 | Q 26 | Page 35

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

Ex. 17.2 | Q 27 | Page 35

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

Ex. 17.2 | Q 28 | Page 35

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

Ex. 17.2 | Q 29 | Page 35

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

Ex. 17.2 | Q 30.1 | Page 35

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

Ex. 17.2 | Q 30.2 | Page 35

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

Ex. 17.2 | Q 31 | Page 35

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) ?

Ex. 17.2 | Q 32 | Page 35

Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?

Ex. 17.2 | Q 33 | Page 35

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).

Ex. 17.2 | Q 34 | Page 35

Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?

Ex. 17.2 | Q 35 | Page 35

Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?

Ex. 17.2 | Q 36 | Page 35

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

Ex. 17.2 | Q 37 | Page 35

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

Ex. 17.2 | Q 38 | Page 35

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] ?

Ex. 17.2 | Q 39.1 | Page 35

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

Ex. 17.2 | Q 39.2 | Page 35

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

Ex. 17.2 | Q 39.3 | Page 35

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]

Q 1 | Page 39

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

Q 2 | Page 39

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

Q 3 | Page 39

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

Q 4 | Page 39

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

Q 5 | Page 39

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

Q 6 | Page 39

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

Q 7 | Page 39

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

Q 8 | Page 40

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

Q 9 | Page 40

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

Q 10 | Page 40

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 ?

Q 11 | Page 40

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

Q 12 | Page 40

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

Q 13 | Page 40

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

Q 14 | Page 40

Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?

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

Q 1 | Page 40

The interval of increase of the function f(x) = x − ex + tan (2π/7) is

  • (0, ∞)

  • (−∞, 0)

  • (1, ∞)

  • (−∞, 1)

Q 2 | Page 40

The function f(x) = cot−1 x + x increases in the interval

  • (1, ∞)

  • (−1, ∞)

  • (−∞, ∞)

  • (0, ∞)

Q 3 | Page 40

The function f(x) = xx decreases on the interval

  • (0, e)

  • (0, 1)

  • (0, 1/e)

  • none of these

Q 4 | Page 40

The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval

  • (1, 2)

  • (2, 3)

  • (1, 3)

  • (2, 4)

Q 5 | Page 40

If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval

  •  (−∞, 4)

  • (4, ∞)

  • (−∞, 8)

  • (8, ∞)

Q 6 | Page 40

Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy

  •  a2 − 3b − 15 > 0

  • a2 − 3b + 15 > 0

  • a2 − 3b + 15 < 0

  • a > 0 and b > 0

Q 7 | Page 40

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

  • even and increasing

  • odd and increasing

  • even and decreasing

  • odd and decreasing

Q 8 | Page 40

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

  • a ∈ (1/2, ∞)

  • a ∈ (−1/2, 1/2)

  • a = 1/2

  • a ∈ R

Q 9 | Page 40

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

  • increasing on (0, π/2)

  • decreasing on (0, π/2)

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

  • none of these

Q 10 | Page 40

Let f(x) = x3 − 6x2 + 15x + 3. Then,

  •  f(x) > 0 for all x ∈ R

  •  f(x) > f(x + 1) for all x ∈ R

  • f(x) is invertible

  • none of these

Q 11 | Page 41

The function f(x) = x2 e−x is monotonic increasing when

  •  x ∈ R − [0, 2]

  • 0 < x < 2

  • 2 < x < ∞

  • x < 0

Q 12 | Page 41

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

  • λ > 1/2

  • λ < 1/2

  • λ < 2

  • λ > 2

Q 13 | Page 41

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

  • monotonically increasing

  • monotonically decreasing

  • not monotonic

  • constant

Q 14 | Page 41

Function f(x) = x3 − 27x + 5 is monotonically increasing when

  • x < −3

  • x | > 3

  • x ≤ −3

  • x | ≥ 3

Q 15 | Page 41

Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when

  •  x < 2

  • x > 2

  •  x > 3

  • 1 < x < 2

Q 16 | Page 41

If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then

  •  k < 3

  • k ≤ 3

  • k > 3

  •  k ≥ 3

Q 17 | Page 41

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

 

  •  x > 0

  • x < 0

  • x ∈ R

  •  x ∈ R − {0}

Q 18 | Page 41

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

 

 

 

 

 

 

 

 

 

 

 

  • x < 0

  •  x > 1

  • x < 1

  • 0 < x < 1

Q 19 | Page 41

Every invertible function is

  • monotonic function

  • constant function

  • identity function

  • not necessarily monotonic function

Q 20 | Page 41

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

  • increasing

  • decreasing

  • constant

  • none of these

Q 21 | Page 41

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

 

  •  a = b

  • \[a = \frac{1}{2}b\]

  • \[a \leq - \frac{1}{2}\]

  • \[a > - \frac{3}{2}\]

Q 22 | Page 41

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

 

  • strictly increasing

  • strictly decreasing

  • neither increasing nor decreasing

  • none of these

Q 23 | Page 41

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

 

  • λ < 1

  • λ > 1

  • λ < 2

  • λ > 2

Q 24 | Page 41

Function f(x) = ax is increasing on R, if

  • a > 0

  • a < 0

  • 0 < a < 1

  • a > 1

Q 25 | Page 41

Function f(x) = loga x is increasing on R, if

  • 0 < a < 1

  • a > 1

  • a < 1

  • a > 0

Q 26 | Page 41

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

  • increases on [0, a]

  • decreases on [0, a]

  • increases on [−a, 0]

  • decreases on [a, 2a]

Q 27 | Page 41

If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then

  •  k ∈ (2, ∞)

  • k ∈ (−∞, 2)

  • k ∈ (4, ∞)

  •  k ∈ (−∞, 4).

Q 28 | Page 41

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

  • increasing

  • decreasing

  • constant

  • none of these

Q 29 | Page 42

If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then

  • −1 ≤ k < 1

  •  k < −1 or k > 1

  • 0 < k < 1

  • −1 < k < 0

Q 30 | Page 42

The function f(x) = x9 + 3x7 + 64 is increasing on

  • R

  • (−∞, 0)

  • (0, ∞)

  •  R0

Chapter 17: Increasing and Decreasing Functions

Ex. 17.1Ex. 17.2Others

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

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

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

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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.

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