Determine the average cost function $\frac{?}{\text{b}\text{a}\text{r}}(C)(x)=\frac{C(x)}{x}.$ To find where the average cost is smallest, first calculate $\frac{?}{\text{b}\text{a}\text{r}}(C{)}^{\text{'}}(x),$ the derivative of the average cost function. Then use a graphing calculator to find where the derivative is $0$ Check your work by finding the minimum from the graph of the function $\frac{?}{\text{b}\text{a}\text{r}}(C)(x)$

$C(x)=\frac{1}{2}{x}^3+5x^2-2x+50$

Determine the average cost function

$\frac{?}{\text{b}\text{a}\text{r}}(C)(x)=\frac{1}{2}{x}^2+5x-2+\frac{50}{x}$

Calculate $\frac{?}{\text{b}\text{a}\text{r}}(C{)}^{\text{'}}(x)$

$\frac{?}{\text{b}\text{a}\text{r}}(C{)}^{\text{'}}(x)=x+5-\frac{50}{x^2}$

$\frac{?}{\text{b}\text{a}\text{r}}(C{)}^{\text{'}}(x)=0$ when $x=$

$($Type an integer or decimal rounded to three decimal places as needed.$)$