Question 76
a) Prove that, if $x, y \ge 1$, then
$$
x + y - \frac{1}{x} - \frac{1}{y} \ge 2\sqrt{xy} - \frac{2}{\sqrt{xy}}.
$$
b) Prove that, if $a, b, c, d \ge 1$ and $abcd = 16$, then
$$
a + b + c + d - \frac{1}{a} - \frac{1}{b} - \frac{1}{c} - \frac{1}{d} \ge 6.
$$
$$
x + y - \frac{1}{x} - \frac{1}{y} \ge 2\sqrt{xy} - \frac{2}{\sqrt{xy}}.
$$
b) Prove that, if $a, b, c, d \ge 1$ and $abcd = 16$, then
$$
a + b + c + d - \frac{1}{a} - \frac{1}{b} - \frac{1}{c} - \frac{1}{d} \ge 6.
$$