\[\boxed{\text{1008.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]
\[a > 0,\ \ b > 0\]
\[\left( \frac{a + b}{2} \right)^{3} \leq \frac{a^{3} + b^{3}}{2}\]
\[\left( \frac{a + b}{2} \right)^{3} \geq \left( \sqrt{\text{ab}} \right)^{3}\]
\[\frac{a³ + b³}{2} \geq \sqrt{a^{3}b^{3}}\]
\[\Longrightarrow \left( \sqrt{\text{ab}} \right)^{3} \leq \sqrt{a^{3}b^{3}}\]
\[\frac{a^{3} + b^{3}}{2} \geq \sqrt{a^{3}b^{3}}\]
\[\text{ab}\sqrt{\text{ab}} \leq ab\sqrt{\text{ab}}\]
\[то\ есть:\ \ \ \ \ \left( \frac{a + b}{2} \right)^{3} \leq\]
\[\leq \frac{a^{3} + b^{3}}{2} \Longrightarrow ч.т.д.\]