Answer
$a)$ = $\frac{\pi^{3}}{192}$
$b)$ = $\frac{\pi^{5}}{46080}$
Work Step by Step
$a)$
$f'(x)$ = -$sinx$
$f''(x)$ = -$cosx$
$f''(0)$ = -$cos(0)$ = -$1$
$f''(\pi)$ = -$cos(\pi)$ = $1$
$max|f''(x)|$ = $1$
$\frac{(b-a)^{3}}{12n^{2}}$$max|f''(x)|$ = $\frac{(\pi)^{3}(1)}{12(4)^{2}}$
= $\frac{\pi^{3}}{192}$
$b)$
$f^{3}(x)$ = $sinx$
$f^{4}(x)$ = $cosx$
$f^{4}(0)$ = $cos(0)$ = $1$
$f^{4}(\pi)$ = $cos(\pi)$ = -$1$
$max|f^{4}(x)|$ = $1$
$\frac{(b-a)^{5}}{180n^{4}}$$max|f^{4}(x)|$ = $\frac{(\pi)^{5}(1)}{180(4)^{4}(1)}$
= $\frac{\pi^{5}}{46080}$
