Single Variable Calculus, Chapter 4, 4.7, Section 4.7, Problem 42

Suppose that $\displaystyle E(\upsilon) = a \upsilon^3 \cdot \frac{L}{\upsilon - u}$. For what value of $\upsilon$ is $E$ smallest? Assuming that other quantities are constant.

$\displaystyle E'(\upsilon) = \frac{(\upsilon - u) (3 a \upsilon^2 L)- (a \upsilon^3 L)(1)}{(\upsilon - u)^2}$
when $E'(\upsilon) = 0$

$
\begin{equation}
\begin{aligned}
0 &= ( \upsilon - u)(3 a\upsilon^2 L) -(a \upsilon^3 L)\\
\\
0 &= a \upsilon^2 L [ ( \upsilon - u) 3 - \upsilon]\\
\\
0 &= (\upsilon - u) 3- \upsilon\\
\\
0 &= 3 \upsilon - 3 u -\upsilon\\
\\
0 &= 2 \upsilon - 3 u\\
\\
\upsilon &= \frac{3u}{2}
\end{aligned}
\end{equation}
$


Therefore, $E$ is smallest when $\displaystyle \upsilon = \frac{3u}{2}$