Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let two numbers be x and y then
x + y = 5 ...(i)
Let S = x3 + y3 ...(ii)
= x3 + (5 – x)3 ...[From (i)]
`(dS)/dx` = 3x2 + 3(5 – x)2 (– 1)
`(dS)/dx` = 3x2 – 3(25 + x2 – 10x)
= 3x2 – 75 – 3x2 + 30x
= 30x – 75
For maximum or minimum
`(dS)/dx` = 0
`\implies` 30x – 75 = 0
`\implies` x = `75/35 = 5/2`
When x = `5/2`, y = `5 - 5/2 = 5/2` ...[From (i)]
`(d^2S)/(dx^2)` = 30 which is +ve.
So the sum is least when x = `5/2` and y = `5/2`
S = x2 + y2
= `25/4 + 25/4`
= `50/4`
= `25/2`
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
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) = sinx − cos x, 0 < x < 2π
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
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.
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.
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?
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
The function f(x) = x log x is minimum at x = ______.
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) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
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`
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The maximum value of `(1/x)^x` is ______.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
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 function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
Divide the number 100 into two parts so that the sum of their squares is minimum.
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
