Advertisements
Advertisements
प्रश्न
Determine the minimum value of the 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
∴ x2 – 7x + 6 = 0
∴ (x – 6)(x – 1) = 0
∴ x – 1 = 0 or x – 6 = 0
∴x = 6 or x = 1
For x = 6,
f''(6) = 12(6) – 42
= 72 – 42
= 30 > 0
∴ f(x) minimum value at x = 6.
f''(1) = 12 – 42
= –30 < 0
∴ f(x) maximum value at x = 1.
f''(6) = 2(6)3 – 21(6)2 + 36(6) – 20
= 432 – 756 + 216 – 20
= –128
∴ The function f(x) has a minimum value of –128 at x = 6.
APPEARS IN
संबंधित प्रश्न
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.
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 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) = x3 − 3x
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) = sinx − cos x, 0 < x < 2π
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
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
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.
If x + y = 3 show that the maximum value of x2y is 4.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
If z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
The function y = 1 + sin x is maximum, when 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.
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/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
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 ______.
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)
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.
If x + y = 8, then the maximum value of x2y is ______.
If \[f\] has a maximum value at \[c\], what is \[c\] called?
For \[f(x)=3x^4+4x^3-12x^2+12\], which is the correct factorization of \[f'(x)\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=1\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=-2\]?
