Inequality (1)

Note that $a^2 \le a^2 + b^2 + c^2 + d^2 \le 4$ , so $a \le 2$ . Hence, $(2 - a)a^2 \ge 0,$ or $a^3 \le 2a^2$ . Similarly, $b^3 \le 2b^2$ , $c^3 \le 2c^2$ , and $d^3 \le 2d^2$ . Adding these up, we get $a^3 + b^3 + c^3 + d^3 \le 2(a^2 + b^2 + c^2 + d^2) = 8.$ Equality occurs when one of $a$ , $b$ , $c$ , $d$ is equal to 2, and the other three are equal to 0.