Advertisements
Advertisements
प्रश्न
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
Advertisements
उत्तर
f(x) = x3 - 9x2 + 24x
∴ f '(x) = 3x2 - 18x + 24
∴ f ''(x) = 6x - 18
Consider, f '(x) = 0
∴ 3x2 - 18x + 24 = 0
∴ 3(x2 - 6x + 8) = 0
∴ 3(x - 4)(x - 2) = 0
∴ (x - 4)(x - 2) = 0
∴ x = 2 or x = 4
For x = 4,
f ''(4) = 6(4) - 18 = 24 - 18 = 6 > 0
∴ f(x) is minimum at x = 4
∴ Minima = f(4) = (4)3 - 9(4)2 + 24(4)
= 64 - 144 + 96 = 16
For x = 2,
f ''(2) = 6(2) - 18 = 12 - 18 = - 6 < 0
∴ f(x) is maximum at x = 2
∴ Maxima = f(2) = (2)3 - 9(2)2 + 24(2) = 8 - 36 + 48 = 20
∴ Maxima = 20 and Minima = 16
APPEARS IN
संबंधित प्रश्न
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
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 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 absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
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.
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^2 + (16)/x^2`
Divide the number 20 into two parts such that sum of their squares is minimum.
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
If x + y = 3 show that the maximum value of x2y is 4.
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
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`
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The maximum value of sin x . cos x is ______.
If y = x3 + x2 + x + 1, then y ____________.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by
f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`
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.
