Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Given function, f(x) = x3 - 6x2 + 9x + 15
`therefore` f'(x) = 3x2 - 12x + 9
= 3 (x2 - 4x + 3)
= 3 (x - 1)(x - 3)
=> x = 1 or x = 3
`therefore` x can have lowest or highest value at x = 1 or x = 3.
f''(x) = 3(2x - 4) = 6x - 12
At, x = 1, f''(x) = 6 × 1 - 12 = - 6 (Negative)
∴ The value of the function at x = 1 is a local maximum.
Maximum value = f(1) = (1)3 - 6(1)2 + 9(1) + 15
= 1 - 6 + 9 + 15
= 19
At x = 3, f'' = 6 × 3 - 12 = 6 positive
∴ f(x) has a local minimum at x = 3.
Minimum value = f(3) = (3)3 - 6(3)2 + 9(3) + 15
= 27 - 54 + 27 + 15
= 15
APPEARS IN
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) = (2x − 1)2 + 3.
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
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:
`h(x) = sinx + cosx, 0 < x < pi/2`
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) = 1/(x^2 + 2)`
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?
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
Find the maximum and minimum of the following functions : f(x) = `logx/x`
Divide the number 30 into two parts such that their product is maximum.
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
Show that among rectangles of given area, the square has least perimeter.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
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.
Solve the following : 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)`.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
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
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
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`
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
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 ____________.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
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)`
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
