suppose c(x)=0.01x^2 + 2x + 3600 is the total cost of a company to produce x units of a certain product…

Explanation:

Step1: Write the average - cost function

Given $C(x)=0.01x^{2}+2x + 3600$, then the average - cost function $A(x)=\frac{C(x)}{x}=\frac{0.01x^{2}+2x + 3600}{x}=0.01x + 2+\frac{3600}{x}$.

Step2: Find the derivative of the average - cost function

Using the power rule, if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. The derivative of $A(x)$ is $A^\prime(x)=0.01-\frac{3600}{x^{2}}$.

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

Set $A^\prime(x) = 0$, so $0.01-\frac{3600}{x^{2}}=0$. Then $\frac{3600}{x^{2}}=0.01$. Cross - multiply to get $0.01x^{2}=3600$. Divide both sides by $0.01$: $x^{2}=\frac{3600}{0.01}=360000$. Take the square root of both sides: $x=\sqrt{360000}=600$ (we consider the positive value of $x$ since $x$ represents the production level).

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

Find the second - derivative $A^{\prime\prime}(x)=\frac{7200}{x^{3}}$. When $x = 600$, $A^{\prime\prime}(600)=\frac{7200}{600^{3}}>0$, so $x = 600$ is a point of minimum.

Answer:

$x = 600.00$