Advertisements
Advertisements
प्रश्न
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
Advertisements
उत्तर
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.
संबंधित प्रश्न
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
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
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.
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening
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
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
Find the maximum and minimum of the following functions : f(x) = x log x
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.
The function f(x) = x log x is minimum at x = ______.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`
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.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
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 ______.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
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 ______.
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 minimum value of the function f(x) = xlogx is ______.
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.
Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is ‘r' cm and height is 'h’ cm. It has a volume of 20π cm3.
- Express ‘h’ in terms of ‘r’, using the given volume.
- Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
- Sumit wants to paint the dustbin. The cost of painting the base and top of the dustbin is ₹ 2 per cm2 and the cost of painting the curved side is ₹ 25 per cm2. Find the total cost in terms of ‘r’, for painting the outer surface of the dustbin including the base and top.
- Calculate the minimum cost for painting the dustbin.
