Question 48
Prove that for any real numbers $a, b$ such that $0\lt a, b\lt1$, the following inequality holds:
$$
\sqrt{a b^{2}+a^{2} b}+\sqrt{(1-a)(1-b)^{2}+(1-a)^{2}(1-b)}\lt\sqrt{2}
$$
$$
\sqrt{a b^{2}+a^{2} b}+\sqrt{(1-a)(1-b)^{2}+(1-a)^{2}(1-b)}\lt\sqrt{2}
$$