Advertisements
Advertisements
प्रश्न
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
Advertisements
उत्तर
Let us consider that the company increases the annual subscription by ₹ x.
So, x is the number of subscribers who discontinue the services.
∴ Total revenue, R(x) = (500 – x)(300 + x)
= 150000 + 500x – 300x – x2
= – x2 + 200x + 150000
Differentiating both sides w.r.t. x,
We get R'(x) = – 2x + 200
For local maxima and local minima, R'(x) = 0
– 2x + 200 = 0
⇒ x = 100
R"(x) = – 2 < 0 Maxima
So, R(x) is maximum at x = 100
Hence, in order to get maximum profit, the company should increase its annual subscription by ₹ 100.
APPEARS IN
संबंधित प्रश्न
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
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) = (x −1)2 + 3, x ∈[−3, 1]
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?
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
Determine the maximum and minimum value of the following function.
f(x) = x log x
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
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
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
If z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
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`
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 maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
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 ______.
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 ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
The minimum value of the function f(x) = xlogx is ______.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
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) `= x sqrt(1 - x), 0 < x < 1`
