suppose $c(x)=0.04x^{2}+4x + 3600$ is the total cost of a company to produce $x$ units of a certain product…

Explanation:

Step1: Find the average - cost function

Given $C(x)=0.04x^{2}+4x + 3600$, then $A(x)=\frac{C(x)}{x}=\frac{0.04x^{2}+4x + 3600}{x}=0.04x + 4+\frac{3600}{x}$.

Step2: Take the derivative of the average - cost function

$A^\prime(x)=\frac{d}{dx}(0.04x + 4+\frac{3600}{x})=0.04-\frac{3600}{x^{2}}$.

Step3: Set the derivative equal to zero and solve for $x$

$0.04-\frac{3600}{x^{2}} = 0$. Add $\frac{3600}{x^{2}}$ to both sides: $0.04=\frac{3600}{x^{2}}$. Cross - multiply: $0.04x^{2}=3600$. Divide both sides by $0.04$: $x^{2}=\frac{3600}{0.04}=90000$. Take the square root of both sides: $x = 300$ (we consider the positive value since $x$ represents the number of units).

Step4: Check the second - derivative to confirm it's a minimum

$A^{\prime\prime}(x)=\frac{d}{dx}(0.04-\frac{3600}{x^{2}})=\frac{7200}{x^{3}}$. When $x = 300$, $A^{\prime\prime}(300)=\frac{7200}{300^{3}}>0$, so $x = 300$ is a point of minimum.

Answer:

$300.00$