Advertisements
Advertisements
Question
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
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
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 f(x) = −(x − 1)2 + 10
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) = x2
Find two numbers whose sum is 24 and whose product is as large as possible.
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
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).`
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 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.
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 : f(x) = `x^2 + (16)/x^2`
Divide the number 30 into two parts such that their product 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.
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.
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.
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals 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 function `f(x) = x^3 - 6x^2 + 9x + 25` has
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
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 ______.
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 ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
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 ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
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.
