Advertisements
Advertisements
प्रश्न
The total cost function of a firm is C = x2 + 75x + 1600 for output x. Find the output for which the average cost ls minimum. Is CA= Cm at this output?
Advertisements
उत्तर
Given C = x2 + 75x + 1600 ..........(1)
Marginal cost Cm = `(dC)/dx`
Differentiating (i) w.r.t.x
Cm = 2x + 75
Average cost CA = `C/x`
= x + 75 + `1600/x`
Diff (ii) w.r.t.x
`(dC_A)/(dx) = 1 + 1600 x ((-1)/x^2)`
= 1 - `1600/x^2`
= `(x^2 - 1600)/x^2`
If `(dC_A)/dx = 0 then .(x^2 - 1600)/x^2 = 0`
`x^2 - 1600 = 0
`x^2 = 1600`
x = 40 and x = -40
Differentiating `(dC_A)/dx` w.r.t.x
`(dC_A)/dx = d/dx (1 - 1600/x^2) = 0 - 1600 xx (-2x^-3)`
`((dC_A)/dx^2) _(at x = 40) = 3200/(40)^3 = 3200/64000`
= `1/20 > 0
Cm = 2x + 75
= 2(40) + 75
= 80 + 75 = 155
CA = x + 75 + `1600/40` = 155
Average cost is minimum for output = 40 .
Since, Cm at this output= 2(40) + .95 = 155
Cm = CA
APPEARS IN
संबंधित प्रश्न
If `y=cos^-1(2xsqrt(1-x^2))`, find dy/dx
Find `dy/dx if y=cos^-1(sqrt(x))`
If y = f (x) is a differentiable function of x such that inverse function x = f –1(y) exists, then
prove that x is a differentiable function of y and
`dx/dy=1/(dy/dx)`, Where `dy/dxne0`
Hence if `y=sin^-1x, -1<=x<=1 , -pi/2<=y<=pi/2`
then show that `dy/dx=1/sqrt(1-x^2)`, where `|x|<1`
Find `dy/dx` if `y = tan^(-1) ((5x+ 1)/(3-x-6x^2))`
If \[y=f(x)\] is a differentiable function of \[x\] and the inverse function \[x=f^{-1}(y)\] exists, which expression gives \[\frac{\mathrm{d}x}{\mathrm{d}y}\]?
The derivative of an inverse function is usually found using which method?
Which expression is the derivative of \[\sin^{-1}(f(x))\] when \[|f(x)|<1\]?
Which expression is the derivative of \[\cos^{-1}(f(x))\] when \[|f(x)|<1\]?
Which derivative and condition are correct for \[\cot^{-1}x\]?
Which expression is the derivative of \[\cot^{-1}(f(x))\] when \[f(x)\in\mathbb{R}\]?
Which derivative and condition are correct for \[\sec^{-1}x\]?
Differentiating \[x=\sin y\] with respect to \[x\] gives which equation?
Using \[\sin^2y+\cos^2y=1\] and \[\sin y=x\], what is \[\cos^2y\]?
For \[y\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\], which value of \[\cos y\] follows from \[\cos^2y=1-x^2\]?
Which inverse functions have denominator \[1+x^2\] in their derivatives?
Which inverse functions have a denominator involving \[|x|\sqrt{x^2-1}\]?
For which inverse functions are negative signs especially important?
