RD Sharma solutions for Mathematics [English] Class 12 chapter 18 - Maxima and Minima [Latest edition]

f(x) = 4x² + 4 on R .

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

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

f(x) = 16x² \[-\] 16x + 28 on R ?

f(x) = x³ \[-\] 1 on R .

f(x) = (x \[-\] 5)⁴.

f(x) = x³ \[-\] 3x.

f(x) = x³  (x \[-\] 1)² .

f(x) =  (x \[-\] 1) (x+2)². 

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

f(x) =  x³ \[-\] 6x² + 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) =\[x\sqrt{1 - x} , x > 0\].

Find the point of local maximum or local minimum, if any, of the following function, using the first derivative test. Also, find the local maximum or local minimum value, as the case may be:

f(x) = x³(2x \[-\] 1)³.

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

f(x) = x⁴ \[-\] 62x² + 120x + 9.

f(x) = x³\[-\] 6x² + 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) = \[- (x - 1 )^3 (x + 1 )^2\] .

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) = x³ + ax² + 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 = \[\frac{\pi}{6}\] ?

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 2x³\[-\] 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) = 2x³ − 15x² + 36x + 1 on the interval [1, 5].

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.

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 \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .

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 \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .

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?

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 m³. If building of tank cost 70 per square metre for the base and Rs 45 per square metre 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 cm³. 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 \[\sqrt{2}\] .

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

Find the coordinates of a point on the parabola y=x²+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?

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?

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 cm². 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.

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) = \[x + \frac{1}{x}, x > 0 .\] 

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

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

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

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

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

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

Let f(x) = x³+3x² \[-\] 9x+2. Then, f(x) has _________________ .

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

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

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

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

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

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

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

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

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

The point on the curve y² = 4x which is nearest to, the point (2,1) is _______________ .

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

The least and greatest values of f(x) = x³\[-\] 6x²+9x in [0,6], are ___________ .

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

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 ______________ .

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

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

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

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

f(x) = 1+2 sin x+3 cos²x, `0<=x<=(2pi)/3` is ________________ .

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

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

Let f(x) = 2x³\[-\] 3x²\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .

The minimum value of x log_e x is equal to ____________ .

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