Problem - 2104
Let $x$ be a real number between 0 and 1. Find the maximum value of $x(1-x^4)$.
\begin{align}
&x(1-x^4)\\
&=\sqrt[4]{x^4(1-x^4)^4} \\
&= \sqrt[4]{\frac{1}{4}\cdot 4x^4(1-x^4)(1-x^4)(1-x^4)(1-x^4)}\\
&\le \sqrt[4]{\frac{1}{4}\cdot\frac{4x^4+(1-x^4)+(1-x^4)+(1-x^4)+(1-x^4)}{5}}\\
& =\boxed{\sqrt[4]{\frac{1}{5}}}
\end{align}