Question

The average cost function, $AC$ for a commodity is given by $AC=x+5+\frac{36}{x},$ in terms of output $x$. Find :
(i). The total cost, $C$ and marginal cost, $MC$ as a function of $x$.
(ii). The outputs for which $AC$ increases.

Answer 1

Given, $AC=x+5+\frac{36}{x}$
(i) $TC=\left(AC\right)x$

$=\left(\begin{array}{c}x+5+\frac{36}{x}\end{array}\right).x$

$=x^2+5x+36$

$MC=\frac{d}{dx}\left(TC\right)$

$=\frac{d}{dx}(x^2+5x+36)$
Ans. $=2x+5$

(ii) AC increases when $\frac{d}{dx}\left(AC\right)>0$
$\frac{d}{dx}\left(\begin{array}{c}x+5+\frac{36}{x}\end{array}\right)=0$

$1-\frac{36}{x^2}>0$

$\Rightarrow x^2-36>0$

$\Rightarrow (x+6)(x-6)>0$

$\Rightarrow x<-6,x>6$
But $x$ is $+ve$ therefore $AC$ increases when $x>6$.