Advertisements
Advertisements
Question
If x + y = 3 show that the maximum value of x2y is 4.
Solution
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
RELATED QUESTIONS
Prove that the following function do not have maxima or minima:
f(x) = ex
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]`
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.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
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 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.
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 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.
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?
Divide the number 20 into two parts such that their product is maximum.
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 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 function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
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 straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
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)
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.
Find the maximum and the minimum values of the function f(x) = x2ex.
