Advertisements
Advertisements
प्रश्न
If x + y = 3 show that the maximum value of x2y is 4.
Advertisements
उत्तर
x + y = 3
∴ y = 3 – x
Let T = x2y = x2(3 – x) = 3x2 – x3
Differentiating w.r.t. x, we get
`"dT"/("d"x) = 6"x" - 3"x"^2` ....(i)
Again, differentiating w.r.t. x, we get
`("d"^2"T")/("d"x^2) = 6 - 6"x"` ...(ii)
Consider, `"dT"/("d"x) = 0`
∴ 6x – 3x2 = 0
∴ x = 2
For x = 2,
`(("d"^2"T")/"dx"^2)_(x = 2)` = 6 – 6(2)
= 6 – 12
= – 6 < 0
Thus, T, i.e., x2y is maximum at x = 2
For x = 2, y = 3 – x = 3 – 2 = 1
∴ Maximum value of T = x2y = (2)2(1) = 4
संबंधित प्रश्न
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
`f(x) = xsqrt(1-x), x > 0`
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]`
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]`
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find two numbers whose sum is 24 and whose product is as large as possible.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
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.
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 profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
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.
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?
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) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
The maximum value of sin x . cos x is ______.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
Divide the number 100 into two parts so that the sum of their squares is minimum.
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
