56 Let $f(x)=a \sin x+b \cos x+c$ be the graph passing through points $A(0,5)$ and $B\left(\frac{\pi}{2}, 5\right)$. When $0 \leqslant x \leqslant \frac{\pi}{2}$, $|f(x)| \leqslant 10$. Find the range of values for $c$.
56 Since the graph of $f(x)$ passes through points $A$ and $B$, we have
$$
b+c=5 \text { and } a+c=5,
$$

thus $a=b=5-c$, hence
$$
f(x)=(5-c) \cdot \sqrt{2} \sin \left(x+\frac{\pi}{4}\right)+c .
$$

Let $\sin \left(x+\frac{\pi}{4}\right)=t$, then $f(x)=g(t)=(5-c) \sqrt{2} t+c$.
$g(t)$ is a linear (or constant) function of $t$ on the interval $\left[\frac{\sqrt{2}}{2}, 1\right]$, so to ensure that $|f(x)| \leqslant 10$ when $0 \leqslant x \leqslant \frac{\pi}{2}$, it is sufficient that $\left|g\left(\frac{\sqrt{2}}{2}\right)\right| \leqslant 10$ and $|g(1)| \leqslant 10$.
Solving this, we get $-5 \sqrt{2} \leqslant c \leqslant 15 \sqrt{2}+20$.