Advertisements
Advertisements
Question
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
Advertisements
Solution
f(x) = `x^2 + 16/x`
∴ f'(x) = `2x - 16/x^2`
and f"(x) = `2 + 32/x^3`
Consider, f'(x) = 0
∴ `2x - 16/x^2` = 0
∴ 2x = `16/x^2`
∴ x3 = 8
∴ x = 2
The maximum value is 2.
f(x) = `x^2 + 16/x`
For x = 2
f''(2) = `2 + 32/2^3`
= `2 + 32/8`
= 2 + 4
= 6 > 0
∴ f(x) attains minimum value at x = 2
∴ Minimum value = f(2) = `(2)^2 + 16/2 = 4 + 8` = 12
∴ The function f(x) has minimum value 12 at x = 2.
RELATED QUESTIONS
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 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) = x3 − 6x2 + 9x + 15
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
It is given that at x = 1, the function x4− 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
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.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
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.
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:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
If f(x) = x.log.x then its maximum value is ______.
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) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 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`
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
The maximum value of sin x . cos x is ______.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 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 ______.
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 2sinx + 2cosx is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
