lean_workbook_plus_5850

Prove that for non-negative real numbers $a, b, c, d, e$, the inequality $b^4 + c^4 + d^4 + e^4 \geq 4bcde$ holds if at least one variable is equal to zero. If all variables are positive, prove that fixing $a^4 + b^4 + c^4 + d^4 + e^4$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e}$ makes the inequality a decreasing function of $abcde$. Show that the inequality is true when four variables are equal to 1 and the fifth variable is $a$ by using AM-GM inequality.