If we let $s(x)=\sin(x)$, and $s^n(x)$ is the $n$'th iteration of $s(x)$, and $$f(x)=s^n(x)-s^{n}(\frac{\pi}{2})s(x)$$For some constant $n$. As we increase $n$ from $0$, the global maximum increases, but then after $n=16$, it decreases again, and will converge to $0$, since $s^{\infty}(x)=0$.
Also, the slope at $x=0$ will increase as $n$ increases, why is the slope increasing and not decreasing like the global maximum?