###### Advertisements

###### Advertisements

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`

###### Advertisements

#### Solution

Let the dimensions of the rectangle be x cm and y cm.

∴ 2x + 2y = 36

∴ x + y = 18

∴ y = 18 – x

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

f(x) = = xy = x(18 – x) = **18x – x ^{2}**

∴ f'(x) = **18 – 2x**

∴ f''(x) = **– 2**

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

18 – 2x = 0

∴ 18 = 2x

∴ x = **9**

∴ f''**(9)** = – 2 < 0

∴ Area is maximum when x = **9**, y = 18 – 9 = **9**

∴ Dimensions of rectangle are **9 cm × 9 cm**. ** **

#### 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

Find the maximum and minimum value, if any, of the following function given by g(x) = x^{3} + 1.

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:

`g(x) = x/2 + 2/x, x > 0`

**Find the absolute maximum value and the absolute minimum value of the following function in the given interval:**

f (x) = sin x + cos x , x ∈ [0, π]

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 positive numbers x and y such that x + y = 60 and xy^{3} is maximum.

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

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?

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`

Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.

Find the points at which the function f given by f (x) = (x – 2)^{4} (x + 1)^{3} has

- local maxima
- local minima
- point of inflexion

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.`

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

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.

Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.

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.

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?

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

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 : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.

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 : 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)`.

Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.

**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______

**State whether the following statement is True or False:**

An absolute maximum must occur at a critical point or at an end point.

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 = ______.

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?

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

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

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`

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

Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______

If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.

If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.

The function y = 1 + sin x is maximum, when x = ______

The maximum and minimum values for the function f(x) = 4x^{3} - 6x^{2} on [-1, 2] are ______

The minimum value of the function f(x) = 13 - 14x + 9x^{2} is ______

The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.

Show that the function f(x) = 4x^{3} – 18x^{2} + 27x – 7 has neither maxima nor minima.

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

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. Also find the maximum volume.

The smallest value of the polynomial x^{3} – 18x^{2} + 96x in [0, 9] is ______.

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

Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`

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}.

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

The distance of that point on y = x^{4} + 3x^{2} + 2x which is nearest to the line y = 2x - 1 is ____________.

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.

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.

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 range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.

Let A = [a_{ij}] be a 3 × 3 matrix, where

a_{ij} = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, "," "otherwise"):}`

Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.

If y = alog|x| + bx^{2} + x has its extremum values at x = –1 and x = 2, then ______.

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 ______.

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 lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, 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 ______.

The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.

A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.

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

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.

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

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 sqrt(1 - x), 0 < x < 1`