The cost function for a company is C(x)=2000+96x+4x^{3/2} where x represents thousands of units. Is there a production level that minimizes average cost?
Average cost is defined by $\displaystyle AC(x)=\frac{C(x)}{x}$.
In order to minimise $\displaystyle AC(x)$ , find $\displaystyle x_0$ such that $\displaystyle AC'(x_0)=0$ and $\displaystyle AC''(x_0)>0$
You will find that $\displaystyle x_0=100$ satisfies these conditions.
$\displaystyle AC(x)=\frac{2000}{x}+96+4x^\frac{1}{2}$
Differentiating wrt x,
$\displaystyle AC'(x)=\frac{-2000}{x^2}+2x^\frac{-1}{2}$
$\displaystyle AC'(x)=0 \Rightarrow 2x^\frac{-1}{2}=\frac{2000}{x^2} \Rightarrow x_0=100$
Similarly find $\displaystyle AC''(x)$ and show that it's value at $\displaystyle x_0=100$ is greater than 0.
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