Advertisements
Advertisements
प्रश्न
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
Advertisements
उत्तर
Given function f(x) = 4x `- 1/2 x^2,` interval `[-2,9/2]`
∴ f'(x) = 4 - `1/2`. 2x = 4 - x
If f'(x) = 0, then 4 - x = 0 ⇒ x = 4
At x = -2, f(-2) = 4 (-2) - `1/2 (-2)^2`
`= - 8 - 1/2 xx 4`
= - 8 - 2
= - 10
At x = 4, `f(4) = 4(4) - (4)^2/2`
`= 16 - 16/2`
= 16 - 8
= 8
At x = `9/2`, `f (9/2) = 4 xx 9/2 - 1/2 xx 81/4`
`= 18 - 81/8`
`= (144 - 81)/8`
`= 63/8`
= 7.875
∴ Absolute maximum value of f (x) = 8 at x = 4
Absolute minimum value of f (x) = -10 at x = -2
APPEARS IN
संबंधित प्रश्न
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
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
Prove that the following function do not have maxima or minima:
g(x) = logx
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
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?
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`
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.`
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.
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) = x log x
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.
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
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)`.
If x + y = 3 show that the maximum value of x2y is 4.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
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) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
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`
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`
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
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 ______.
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?
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... 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 ______.
Find the maximum and the minimum values of the function f(x) = x2ex.
A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.
Divide the number 100 into two parts so that the sum of their squares is minimum.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
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
