Algebra At Christmas – Mattia Giuri's bizarre blog

Why not? I mean, it’s a nice inequality to try out. What if I asked you to find the maximum value of

$x(1-x^3), x \in [0, 1]$

Well, time to apply the AM-GM inequality. First, let’s say that $a = x(1-x^3)$

Notice that

$3a^3 \leq (\frac{3x^3+3(1-x^3)}{4})^4 = \frac{81}{256}$

by the AM-GM inequality applied on the 4 positive real numbers $3x^3, 1-x^3, 1-x^3, 1-x^3$

Hence, we get that

$a \leq \frac{3}{4^{\frac{4}{3}}}$

with equality when $3x^3 = 1-x^3 \Rightarrow x = \frac{1}{4^{\frac{1}{3}}}$