We have the next cost given function:
[tex]C=2.5x^2+75x+25000[/tex]
The average cost function is given by:
[tex]A(x)=\frac{C(x)}{x}[/tex]
Replace:
[tex]A(x)=\frac{2.5x^2+75x+25000}{x}[/tex]
Simplify the expression:
[tex]\begin{gathered} A(x)=\frac{2.5x^2}{x}+\frac{75x}{x}+\frac{25000}{x} \\ \text{Then:} \\ A(x)=2.5x+75+\frac{25000}{x} \end{gathered}[/tex]
Derivate A(x)
[tex]A^{\prime}(x)=2.5(1)+0-\frac{25000}{x^2}[/tex]
Set A'(x)=0
[tex]0=2.5(1)+0-\frac{25000}{x^2}[/tex]
Solve for x:
[tex]\begin{gathered} 0=2.5-\frac{25000}{x^2} \\ -2.5=-\frac{25000}{x^2} \\ x^2=\frac{-25000}{-2.5} \\ x^2=10000 \\ \text{Take square root of both sides:} \\ \sqrt[]{x^2}=\sqrt[]{10000} \\ \text{Hence } \\ x=100 \end{gathered}[/tex]
Therefore, the average cost per unit will be minimized at 100 units of production level.
