Advertisements
Advertisements
Question
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
Advertisements
Solution
f(x) = 2x3 – 21x2 + 36x – 20
∴ f'(x) = 6x2 – 42x + 36 and f''(x) = 12x – 42
Consider, f '(x) = 0
∴ 6x2 – 42x + 36 = 0
∴ 6(x2 – 7x + 6) = 0
∴ 6(x – 1)(x - 6) = 0
∴ (x – 1)(x – 6) = 0
∴ x – 1 = 0 or x – 6 = 0
∴ x = 1 or x = 6
For x = 1,
f''(1) = 12(1) – 42 = 12 – 42 = – 30 < 0
∴ f(x) attains maximum value at x = 1.
∴ Maximum value = f(1)
= 2(1)3 – 21(1)2 + 36(1) – 20
= 2 – 21 + 36 – 20
= – 19 – 20 + 36
= – 39 + 36
= – 3
∴ The function f(x) has maximum value – 3 at x = 1.
For x = 6,
f''(6) = 12(6) – 42 = 72 – 42 = 30 > 0
∴ f(x) attains minimum value at x = 6.
∴ Minimum value = f(6)
= 2(6)3 – 21(6)2 + 36(6) – 20
= 432 – 756 + 216 – 20
= – 128
∴ The function f(x) has minimum value – 128 at x = 6.
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 the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box
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?
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
Divide the number 20 into two parts such that sum of their squares is minimum.
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.
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:
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.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
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 `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
If y = x3 + x2 + x + 1, then y ____________.
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
Range of projectile will be maximum when angle of projectile is
A function f(x) is maximum at x = a when f'(a) > 0.
Let A = [aij] be a 3 × 3 matrix, where
aij = `{{:(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 ______.
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b 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 ______.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
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`
What is an absolute maximum?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=-2\]?
