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
Advertisements
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
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:
h(x) = x3 + x2 + x + 1
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
What is the maximum value of the function sin x + cos x?
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
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
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.
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
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.
The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
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.
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.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
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?
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
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?
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
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 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`
An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` 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?
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.
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 f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, 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.
Complete the following activity to divide 84 into two parts such that the product of one part and square of the other is maximum.
Solution: Let one part be x. Then the other part is 84 - x
Letf (x) = x2 (84 - x) = 84x2 - x3
∴ f'(x) = `square`
and f''(x) = `square`
For extreme values, f'(x) = 0
∴ x = `square "or" square`
f(x) attains maximum at x = `square`
Hence, the two parts of 84 are 56 and 28.
If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
