How to solve these inequations I would appreciate if somebody could help me with the following problem:
Q: show that  (  $x\in \left[0,\frac{\pi}{2} \right], a,b>0$)
$$a+b\leq \sqrt{a^2 \sin^2 x+b^2 \cos^2x}+\sqrt{a^2 \cos^2x+b^2 \sin^2x}\leq \sqrt{2} \sqrt{a^2+b^2}$$

Hint: Using $(x+y)^2 \le 2(x^2+ y^2)$ to show the right inequality.
For the left, squaring and canceling the same terms we get
$$ab\le \sqrt{(a^2\sin^2x+b^2\cos^2x)(a^2\cos^2x + b^2\sin^2x)},$$
the rest is yours.
