#### 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

## Chapter 18: Maxima and Minima

### RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.1 [Page 7]

f(x) = 4x^{2} + 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 .

### RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.2 [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)=sin2x-x, -pi/2<=x<=pi/2`

`f(x)=2sinx-x, -pi/2<=x<=pi/2`

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

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

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

### RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.3 [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>=-2` .

`f(x)=xsqrt(32-x^2), -5<=x<=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)=xsqrt(1-x), x<=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 = \[\frac{\pi}{6}\] ?

### RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.4 [Page 37]

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

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

`f(x) = 3x^4 - 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 \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?

### RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.5 [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 a given 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 a given 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 sphere 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 x^{2} + y^{2} =r^{2}, 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 y^{2} = 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. \[\left( 50 - \frac{x}{2} \right) .\] Find 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.

### RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima [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 of `f' (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 .

### RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima [Pages 80 - 82]

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

`e^(1/e)`

`(1/e)^e`

1

none of these

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

`ab<c^2/4`

`ab>=c^2/4`

`ab>=c/4`

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

e

1/e

1

none of these

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

x = 1 is a point of maximum

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

maximum value > minimum value

maximum value < minimum value

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

a maximum at x = 1

a minimum at x = 1

neither a maximum nor a minimum at x = - 3

none of these

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

6

4

8

none of these

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

\[\frac{1}{2}\]

\[\frac{1}{4}\]

\[\frac{3}{4}\]

none of these

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

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

\[\sqrt[3]{abc}\]

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

none of these

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

\[\frac{1}{4}\]

\[\frac{1}{2}\]

\[\frac{1}{8}\]

none of these

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

5

`5/2`

3

2

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

0

maximum

minimum

none of these

If x lies in the interval [0,1], then the least value of x^{2} + x + 1 is _______________ .

3

`3/4`

1

none of these

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

126

135

160

0

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

\[ \frac{1}{4}\]

\[- \frac{1}{3}\]

\[\frac{1}{6}\]

\[\frac{1}{5}\]

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

\[1, 2\sqrt{2}\]

(1, 2)

(1, -2)

( -2,1)

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

8

16

20

24

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

3, 4

0, 6

0, 3

3, 6

0, 54

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

\[\frac{\pi}{3}\]

\[\frac{\pi}{4}\]

\[\frac{\pi}{6}\]

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 ______________ .

\[\frac{3}{4}\]

\[\frac{1}{3}\]

\[\frac{1}{4}\]

\[\frac{2}{3}\]

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

75

50

25

55

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

-2

0

3

none of these

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

\[\frac{4}{3}\]

\[\frac{2}{3}\]

1

\[\frac{3}{4}\]

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

1

2

\[2\frac{1}{2}\]

\[3\frac{1}{3}\]

f(x) = 1+2 sin x+3 cos^{2}x, `0<=x<=(2pi)/3` is ________________ .

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

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

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

Maximum at `sin^-1(1/6)`

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

3

0

4

2

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

\[- \frac{1}{4}\]

\[- \frac{1}{3}\]

\[\frac{1}{6}\]

\[\frac{1}{5}\]

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

-2

-1

2

4

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

e

`1/e`

`-1/e`

`2/e`

`-e`

The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .

-128

-126

-120

none of these

## Chapter 18: Maxima and Minima

## RD Sharma solutions for Class 12 Maths chapter 18 - Maxima and Minima

RD Sharma solutions for Class 12 Maths chapter 18 (Maxima and Minima) 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 Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Maths chapter 18 Maxima and Minima 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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