Problem

Let $f(x)=\sin(x)/x$ for $x\ne 0$ and $g(x)=x\cos(x)-\sin(x)$ for $x\in(-\infty,\infty)$.

(a) (5 pts) Is it possible to define $f(0)$ so that $f$ is differentiable at $0$? Justify your answer.

(b) (5 pts) Suppose that $I$ is an open interval and $I\subset(0,\infty)$. Show that if $g(x)>0$ for $x\in I$, then $f$ is increasing on $I$.

(c) (5 pts) Show that $g$ is decreasing on $(0,\pi/2)$.

(d) (5 pts) Suppose that $f(0)$ is defined so that $f$ is continuous at $0$. Does $f$ have a relative maximum at $0$? Justify your answer.