Question 65
Let $m \ge 0$ and $f(x) = x^2 + \sqrt{mx} + m + 1$. Prove that for all positive real numbers $x_1, x_2, \dots, x_n$,
$$
f(\sqrt[n]{x_1 \cdots x_n}) \le \sqrt[n]{f(x_1)f(x_2)\cdots f(x_n)},
$$
with equality if and only if $x_1 = x_2 = \cdots = x_n$.
$$
f(\sqrt[n]{x_1 \cdots x_n}) \le \sqrt[n]{f(x_1)f(x_2)\cdots f(x_n)},
$$
with equality if and only if $x_1 = x_2 = \cdots = x_n$.