Advertisements
Advertisements
प्रश्न
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
Advertisements
उत्तर
For maxima and minima,
P'(x) = 0
`\implies` 42 – 2x = 0
`\implies` x = 21 and P"(x) = –2 < 0
So, P(x) is maximum at x = 21.
The maximum value of P(x) = 72 + (42 × 21) – (21)2 = 513
i.e., the maximum profit is ₹ 513.
APPEARS IN
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Prove that the following function do not have maxima or minima:
g(x) = logx
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
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.
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:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
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
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
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.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
Read the following passage and answer the questions given below.
|
In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1. |
- If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
- Find the critical point of the function.
- Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
OR
Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by
f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`
The minimum value of 2sinx + 2cosx is ______.
The minimum value of the function f(x) = xlogx 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)
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
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?
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
