Advertisements
Advertisements
प्रश्न
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
पर्याय
`(1/3)^(1/3)`
`1/2`
1
0
उत्तर
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is 1.
Explanation:
Let, `y = [x (x – 1) + 1]^(1/3)`
Differentiating both sides with respect to x,
`dy/dx = 1/3 [x (x - 1) + 1]^(-2/3) d/dx [x(x - 1) + 1]`
`= 1/3 [x (x - 1) + 1]^(-2/3) × (2x - 1)`
`= (2x - 1)/(3 [x (x - 1) + 1]^(2/3))`
For highest and lowest value, `dy/dx = 0 => 2x - 1 = 0 => x = 1/2`
For highest and lowest value, `dy/dx = 0 => 2x - 1 = 0 => x = 1/2`
At `x= 0, f(0) = 1^(1/3) = 1`
At `x= 1, f(1) = 1^(1/3) = 1`
x `= 1/2 at, f(1/2) = [1/2 (-1/2) xx 1]^(1/3) = (3/4)^(1/3)`
`dy/dx , at x = 1/2` sign is changing from -ve to +ve
∴ y is minimum at x = `1/2`.
maximum value = 1
APPEARS IN
संबंधित प्रश्न
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
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) = −(x − 1)2 + 10
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 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 and minimum values of x + sin 2x on [0, 2π].
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.
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?
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening
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?
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) = 2x3 – 21x2 + 36x – 20
Divide the number 20 into two parts such that sum of their squares is minimum.
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
A metal wire of 36cm long is bent to form a rectangle. Find it's dimensions when it's area is maximum.
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
The function y = 1 + sin x is maximum, when x = ______
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
A function f(x) is maximum at x = a when f'(a) > 0.
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 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 minimum value of the function f(x) = xlogx is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
