Problem

Suppose that $f$ is a differentiable function such that $f'(x)=-f(x)$ and $f(x)>0$ for all $x\in(-\infty,\infty)$. Which of the following statement is false?

(a) $f(x)=f(0)e^{-x}$ for $x\in(-\infty,\infty)$.

(b) $f''(x)=f(x)$ for $x\in(-\infty,\infty)$.

(c) $f$ is strictly increasing on $(-\infty,\infty)$.

(d) Let $g(x)=\dfrac{d}{dx}\ln(f(x))$ for $x\in(-\infty,\infty)$. Then $g$ is a constant function on $(-\infty,\infty)$.

(e) $\lim_{x\to\infty}f(x)=0$.