###### Advertisements

###### Advertisements

Divide the number 20 into two parts such that their product is maximum

###### Advertisements

#### Solution

The given number is 20.

Let x be one part of the number and y be the other part.

∴ x + y = 20

∴ y = (20 – x) ......(i)

The product of two numbers is xy.

∴ f(x) = xy

= x(20 – x) ......[From (i)]

= 20x – x^{2}

∴ f'(x) = 20 – 2x and f''(x) = – 2

Consider, f'(x) = 0

∴ 20 – 2x = 0

∴ 20 = 2x

∴ x = 10

For x = 10,

f''(10) = – 2 < 0

∴ f(x), i.e., product is maximum at x = 10

and y = 20 – 10 ......[From (i)]

i.e., y = 10

#### APPEARS IN

#### RELATED QUESTIONS

Examine the maxima and minima of the function f(x) = 2x^{3} - 21x^{2} + 36x - 20 . Also, find the maximum and minimum values of f(x).

If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).

Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]

Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.

An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.

If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius *r* is `(4r)/3`. Also find maximum volume in terms of volume of the sphere

A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.

Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.

Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:

f(x) = sinx − cos x, 0 < x < 2π

Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:

f(x) = x^{3} − 6x^{2}^{ }+ 9x + 15

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

`f(x) = xsqrt(1-x), x > 0`

**Prove that the following function do not have maxima or minima:**

g(x) = logx

**Prove that the following function do not have maxima or minima:**

h(x) = x^{3} + x^{2} + x + 1

What is the maximum value of the function sin x + cos x?

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 two numbers whose sum is 24 and whose product is as large as possible.

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

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner 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 the maximum possible?

Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.

The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.

Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?

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

A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.

The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.

Find the point on the straight line 2x+3y = 6, which is closest to the origin.

A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.

Find the maximum and minimum of the following functions : y = 5x^{3} + 2x^{2} – 3x.

Find the maximum and minimum of the following functions : f(x) = 2x^{3} – 21x^{2} + 36x – 20

Find the maximum and minimum of the following functions : f(x) = x^{3} – 9x^{2} + 24x

Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`

Find the maximum and minimum of the following functions : f(x) = `logx/x`

Divide the number 30 into two parts such that their product is maximum.

Divide the number 20 into two parts such that sum of their squares is minimum.

A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.

A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t^{2} . Find the maximum height it can reach.

Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.

An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.

The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?

A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?

The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.

Show that among rectangles of given area, the square has least perimeter.

Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.

Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.

**Choose the correct option from the given alternatives : **

If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.

Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a^{2}. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.

Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.

Solve the following : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.

Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.

**Solve the following: **

A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.

**Solve the following: **

A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.

**Solve the following: **

Find the maximum and minimum values of the function f(x) = cos^{2}x + sinx.

**Determine the maximum and minimum value of the following function.**

f(x) = `x^2 + 16/x`

Divide the number 20 into two parts such that their product is maximum.

A metal wire of 36cm long is bent to form a rectangle. Find it's dimensions when it's area is maximum.

The total cost of producing x units is ₹ (x^{2} + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?

**Fill in the blank:**

If f(x) = x log x, then its minimum value is______

If x + y = 3 show that the maximum value of x^{2}y is 4.

Examine the function for maxima and minima f(x) = x^{3} - 9x^{2} + 24x

The function f(x) = x log x is minimum at x = ______.

Find the local maximum and local minimum value of f(x) = x^{3} − 3x^{2} − 24x + 5

A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum

A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?

A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.

A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.

**Solution:** Let the dimensions of the rectangle be x cm and y cm.

∴ 2x + 2y = 36

Let f(x) be the area of rectangle in terms of x, then

f(x) = `square`

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme value, f'(x) = 0, we get

x = `square`

∴ f''`(square)` = – 2 < 0

∴ Area is maximum when x = `square`, y = `square`

∴ Dimensions of rectangle are `square`

By completing the following activity, examine the function f(x) = x^{3} – 9x^{2} + 24x for maxima and minima

**Solution:** f(x) = x^{3} – 9x^{2} + 24x

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme values, f'(x) = 0, we get

x = `square` or `square`

∴ f''`(square)` = – 6 < 0

∴ f(x) is maximum at x = 2.

∴ Maximum value = `square`

∴ f''`(square)` = 6 > 0

∴ f(x) is maximum at x = 4.

∴ Minimum value = `square`

If f(x) = px^{5} + qx^{4} + 5x^{3} - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.

The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.

The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.

If f(x) = 3x^{3} - 9x^{2} - 27x + 15, then the maximum value of f(x) is _______.

The maximum value of function x^{3} - 15x^{2} + 72x + 19 in the interval [1, 10] is ______.

The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.

An open box with square base is to be made of a given quantity of cardboard of area c^{2}. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units

A metal box with a square base and vertical sides is to contain 1024 cm^{3}. The material for the top and bottom costs Rs 5/cm^{2} and the material for the sides costs Rs 2.50/cm^{2}. Find the least cost of the box.

Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`

Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x^{2}.

The function f(x) = x^{5} - 5x^{4} + 5x^{3} - 1 has ____________.

Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.

The coordinates of the point on the parabola y^{2} = 8x which is at minimum distance from the circle x^{2} + (y + 6)^{2} = 1 are ____________.

The combined resistance R of two resistors R_{1} and R_{2} (R_{1}, R_{2} > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R_{1} + R_{2} = C (a constant), then maximum resistance R is obtained if ____________.

#### Let f(x) = 1 + 2x^{2} + 2^{2}x^{4} + …… + 2^{10}x^{20}. Then f (x) has ____________.

A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second^{2})

Range of projectile will be maximum when angle of projectile is

The function `f(x) = x^3 - 6x^2 + 9x + 25` has

The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is

For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`

The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is

The maximum value of the function f(x) = `logx/x` is ______.

Divide 20 into two ports, so that their product is maximum.

A function f(x) is maximum at x = a when f'(a) > 0.

Let f: R → R be a function defined by f(x) = (x – 3)^{n1}(x – 5)^{n2}, n_{1}, n_{2} ∈ N. Then, which of the following is NOT true?

The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.

Let P(h, k) be a point on the curve y = x^{2} + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.

If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.

The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.

Let x and y be real numbers satisfying the equation x^{2} – 4x + y^{2} + 3 = 0. If the maximum and minimum values of x^{2} + y^{2} are a and b respectively. Then the numerical value of a – b is ______.

The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.

The minimum value of 2^{sinx} + 2^{cosx} is ______.

A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle is ______.

The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.

The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.

A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.

Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.

**Read the following passage:**

Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
^{2}. |

**Based on the above information, answer the following questions:**

- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
**OR**

(b) For maximum volume, h > r. State true or false and justify. (2)

Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.

Check whether the function f : R `rightarrow` R defined by f(x) = x^{3} + x, has any critical point/s or not ? If yes, then find the point/s.

A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.

Complete the following activity to divide 84 into two parts such that the product of one part and square of the other is maximum.

**Solution:** Let one part be x. Then the other part is 84 - x

Letf (x) = x^{2} (84 - x) = 84x^{2} - x^{3}

∴ f'(x) = `square`

and f''(x) = `square`

For extreme values, f'(x) = 0

∴ x = `square "or" square`

f(x) attains maximum at x = `square`

Hence, the two parts of 84 are 56 and 28.

If Mr. Rane order x chairs at the price p = (2x^{2} - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?

**Solution: **Let Mr. Rane order x chairs.

Then the total price of x chairs = p·x = (2x^{2} - 12x- 192)x

= 2x^{3 }- 12x^{2} - 192x

Let f(x) = 2x^{3} - 12x^{2} - 192x

∴ f'(x) = `square` and f''(x) = `square`

f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0

∴ f is minimum when x = 8

Hence, Mr. Rane should order 8 chairs for minimum cost of deal.

The rectangle has area of 50 cm^{2}. Complete the following activity to find its dimensions for least perimeter.

**Solution: **Let x cm and y cm be the length and breadth of a rectangle.

Then its area is xy = 50

∴ `y =50/x`

Perimeter of rectangle `=2(x+y)=2(x+50/x)`

Let f(x) `=2(x+50/x)`

Then f'(x) = `square` and f''(x) = `square`

Now,f'(x) = 0, if x = `square`

But x is not negative.

∴ `x = root(5)(2) "and" f^('')(root(5)(2))=square>0`

∴ by the second derivative test f is minimum at x = `root(5)(2)`

When x = `root(5)(2),y=50/root(5)(2)=root(5)(2)`

∴ `x=root(5)(2) "cm" , y = root(5)(2) "cm"`

Hence, rectangle is a square of side `root(5)(2) "cm"`

Find the maximum and the minimum values of the function f(x) = x^{2}e^{x}.

A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.

A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.

If x + y = 8, then the maximum value of x^{2}y is ______.

Divide the number 100 into two parts so that the sum of their squares is minimum.

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

A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?

Determine the minimum value of the function.

f(x) = 2x^{3} – 21x^{2} + 36x – 20