suppose (c(x)=0.03x^{2}+2x + 4100) is the total cost of a company to produce (x) units of a certain product…

Explanation:

Step1: Find the average - cost function

Given $C(x)=0.03x^{2}+2x + 4100$, then $A(x)=\frac{C(x)}{x}=\frac{0.03x^{2}+2x + 4100}{x}=0.03x + 2+\frac{4100}{x}$.

Step2: Take 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 $A^\prime(x)=0.03-\frac{4100}{x^{2}}$.

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

Set $A^\prime(x) = 0$, so $0.03-\frac{4100}{x^{2}}=0$. Then $\frac{4100}{x^{2}}=0.03$. Cross - multiply to get $0.03x^{2}=4100$. So $x^{2}=\frac{4100}{0.03}=\frac{410000}{3}$. Then $x=\sqrt{\frac{410000}{3}}\approx369.37$ (we take the positive value since $x$ represents the production level).

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

The second - derivative $A^{\prime\prime}(x)=\frac{8200}{x^{3}}$. When $x = \sqrt{\frac{410000}{3}}$, $A^{\prime\prime}(x)>0$, which means $A(x)$ has a minimum at this $x$ value.

Answer:

$369.37$