Advertisements
Advertisements
प्रश्न
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
Advertisements
उत्तर
\[\text { Profit =S.P. - C.P}.\]
\[ \Rightarrow P = x\left( 50 - \frac{x}{2} \right) - \left( \frac{x^2}{4} + 35x + 25 \right)\]
\[ \Rightarrow P = 50x - \frac{x^2}{2} - \frac{x^2}{4} - 35x - 25\]
\[ \Rightarrow \frac{dP}{dx} = 50 - x - \frac{x}{2} - 35\]
\[\text { For maximum or minimum values of P, we must have }\]
\[\frac{dP}{dx} = 0\]
\[ \Rightarrow 15 - \frac{3x}{2} = 0\]
\[ \Rightarrow 15 = \frac{3x}{2}\]
\[ \Rightarrow x = \frac{30}{3}\]
\[ \Rightarrow x = 10\]
\[\text { Now,} \]
\[\frac{d^2 P}{d x^2} = \frac{- 3}{2} < 0\]
\[\text{ So, profit is maximum if daily output is 10 items.}\]
APPEARS IN
संबंधित प्रश्न
f(x) = 16x2 \[-\] 16x + 28 on R ?
f(x) = x3 \[-\] 3x.
f(x) = x3 (x \[-\] 1)2 .
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
Find the absolute maximum and minimum values of a function f given by `f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 1]` .
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners 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 maximum possible?
A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.
Prove that a conical tent of given capacity will require the least amount of canavas when the height is \[\sqrt{2}\] times the radius of the base.
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.
A straight line is drawn through a given point P(1,4). Determine the least value of the sum of the intercepts on the coordinate axes ?
The space s described in time t by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the minimum value of f(x) = xx .
For the function f(x) = \[x + \frac{1}{x}\]
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
