Putnam 1987 A3

For all real $x$, the real-valued function $y=f(x)$ satisfies $$y''-2y'+y=2e^x.$$

(a) If $f(x)>0$ for all real $x$, must $f'(x) > 0$ for all real $x$? Explain.

(b) If $f'(x)>0$ for all real $x$, must $f(x) > 0$ for all real $x$? Explain.