Advertisements
Advertisements
प्रश्न
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]`
Advertisements
उत्तर
f (x) =x3, x ∈ [-2, 2]
= f' (x) = 3x2
For critical points, f' (x) = 0
= 3x2 = 0
= x = 0 ∈ [-2, 2]
Hence, for finding the absolute maximum value and the absolute minimum value, we have to evaluate f (0), f (-2) and f (2).
Now f(0) = 03, f(-2) = (-2)3 = -8 and f (2) = 23 = 8
∴ Absolute maximum value of f (x) = 8 at x = 2 and absolute minimum value of f (x) = -8 at x = -2.
APPEARS IN
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 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:
`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) = x/2 + 2/x, x > 0`
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]
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.
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.
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
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?
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
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 rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
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.
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
Solve the following : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
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 maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
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?
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
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.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
Divide the number 100 into two parts so that the sum of their squares is minimum.
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
