Problem

Suppose that $f(x)=x^4-2x^3+1$ for $x\in(-\infty,\infty)$ and $g$ is a function defined on $(-\infty,\infty)$ such that $g'(0)=1$. Let $h(x)=f(x)g(x)$ for $x\in(-\infty,\infty)$. Which of the following statements is true?

(a) $-\infty<h'(0)\le 0$.

(b) $0<h'(0)\le 1$.

(c) $1<h'(0)<\infty$.

(d) $h'(0)$ exists but the value of $h'(0)$ cannot be determined based on the given information.

(e) The existence of $h'(0)$ cannot be guaranteed based on the given information.