Advertisements
Advertisements
प्रश्न
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
Advertisements
उत्तर
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.
संबंधित प्रश्न
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
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.
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
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) = 1/(x^2 + 2)`
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:
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 x + y = 60 and xy3 is maximum.
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
Divide the number 30 into two parts such that their product is maximum.
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
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?
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)`.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
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.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
If y = x3 + x2 + x + 1, then y ____________.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
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 second2)
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
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.
|
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)
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
