1. Find the maximum value of the function $f(x)=(\sin x)^{\cos x}$ on the interval $\left[0, \frac{\pi}{2}\right]$.
1. On the observed interval, $0 \leqslant \sin x \leqslant 1$ and $0 \leqslant \cos x \leqslant 1$ hold, so $(\sin x)^{\cos x} \leqslant 1^{\cos x}=1$. This value is indeed achieved for $x=\frac{\pi}{2}$, so the desired maximum value is 1.