Answer
Relative minimum: $(0, -1)$
Relative maxima: $(\pm\pi, 11.592)$
Work Step by Step
$f(x)=\sin x\sinh x-\cos x\cosh x,\ -4\leq x\leq 4$
$f^{\prime}(x)=\sin x\cosh x+\cos x\sinh x-\cos x\sinh x+\sin x\cosh x$
$=2\sin x\cosh x$
$f^{\prime}(x)=0 $ on $-4\leq x\leq 4$ when
$\sin x=0, \quad $or $\cosh x=0$
$x=\pm\pi, \quad $or$ x=0$
Second derivative test:
$f^{\prime\prime}(x)=2(\cos x\cosh x+\sin x\sinh x)\\$
$ f^{\prime\prime}(0)=2(1\cdot 1+0) > 0,\qquad$ min
$ f^{\prime\prime}(\pm\pi)=2(-1\cdot\cosh\pi+0) < 0 , \qquad$ max
