Advertisements
Advertisements
Question
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
Advertisements
Solution
Given function f(x) = (x - 1)2 + 3 in the interval [-3, 1]
∴ f'(x) = 2(x - 1)
For critical points, let f' (x) = 0
If f'(x) = 0, then 2(x - 1) = 0,
⇒ x = 1 ∈ [-3, 1]
At, x = 1 f(1) = (1 - 1)2 + 3
= 0 + 3
= 3
At, x = -3 f(-3)
= (-3, -1)2 + 3
= 16 + 3 = 19
∴ Absolute maximum value of f(x) 19 at x = -3
Absolute minimum value of f(x) = 3 at x = 1.
APPEARS IN
RELATED QUESTIONS
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
Find the maximum and minimum value, if any, of the following function given by g(x) = − |x + 1| + 3.
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:
`g(x) = 1/(x^2 + 2)`
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
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].
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).`
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.
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.`
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
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.
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
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.
The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
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 x is real, the minimum value of x2 – 8x + 17 is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of `(1/x)^x` is ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
Divide 20 into two ports, so that their product is maximum.
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 maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 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).
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
The shortest distance between the line y - x = 1and the curve x = y2 is
