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3 problems match “monotonicity”. Clear

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1. Number of solutions of a trig-root equation ●●○○○ 20 ptsFind the number of real solutions $x$ of the equation $\sin\left(\frac{\pi\sqrt x}{4}\right)+\cos\left(\frac{\pi\sqrt{2-x}}{4}\right)=\sqrt 2$.
2. Sum of x and y in a system ●●●○○ 50 ptsGiven that $x+\sin y=2008$ and $x+2008 \cos y=2007$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$. Give your answer as a decimal rounded to two decimal places.
3. Maximum integer k in a derivative inequality ●●●●○ 100 ptsLet $f(x)=e^{x}-x-2$. For $x>0$, the inequality $(x-k)f'(x)+x+1>0$ always holds, where $k$ is an integer. Find the maximum value of $k$.