= Solution
Differentiating gives
$$
y'=e^{\sin^2x}\sin2x,\qquad
y''=e^{\sin^2x}\left(\sin^22x+2\cos2x\right).
$$
The exponential never vanishes, so all stationary points have $x=k\pi/2$. At $x=k\pi$, $y=1$ and $y''=2>0$; at $x=\pi/2+k\pi$, $y=e$ and $y''=-2e<0$. Thus
$$
\boxed{(k\pi,1)\text{ are minima},\qquad
(\pi/2+k\pi,e)\text{ are maxima},\quad k\in\mathbb Z}.
$$
These are <global minima> and maxima because $0\leq\sin^2x\leq1$. None of the stationary points is an <inflection point>.
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