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
Find the maximum and minimum value, if any, of the following function given by f(x) = (2x − 1)2 + 3.
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
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 positive numbers x and y such that x + y = 60 and xy3 is maximum.
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
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 given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
Find the maximum and minimum of the following functions : f(x) = x log x
Find the maximum and minimum of the following functions : f(x) = `logx/x`
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.
Show that among rectangles of given area, the square has least perimeter.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
The function y = 1 + sin x is maximum, when x = ______
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
A function f(x) is maximum at x = a when f'(a) > 0.
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 p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
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 ______.
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)
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are
