Let $f:\left [ 0,\frac{\pi }{2} \right ]\rightarrow \mathbb{R}, f(x)=\sin(\cos(x))+\cos(\sin(x))$. Prove that $$\int\limits_{0}^{\pi/2}f(x)\,\mathrm{d}x\leq \frac{\pi ^{2}}{4}\,.$$
I've managed to find that $f$ is decreasing on $\left [ 0,\frac{\pi }{2} \right ]$, hence: $$ \int\limits_{0}^{\pi/2} f(x)\,\mathrm{d}x \leq \int\limits_{0}^{\pi/2} f(0)\,\mathrm{d}x = \frac{\pi }{2}\cdot (1+\sin(1))$$ But $1+\sin(1)\geq \frac{\pi }{2}$, thus my inequality isn't enough to prove the problem statement.