Let $x \in [0,1]$. How can I show the inequality $x^p - x^{p+1} \leq \frac{1}{p+1}$?
2026-04-06 17:51:46.1775497906
Proof of the inequality $x^p - x^{p+1} \leq \frac{1}{p+1}$ for $p \in [0,1]$
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Hint: We have by the Weighted AM-GM Inequality that $$x^p(p-px)\leq \left(\frac{px+(p-px)}{p+1}\right)^{p+1}\leq \frac{p}{p+1}\,.$$ (Here, $p$ can be any positive real number.)