Problem

Let

$$f(x)=x-\int_0^{x^2}e^{-t^2}\,dt$$

for $x\in(-\infty,\infty)$. Which of the following statements is true?

(a) $f$ is strictly decreasing on the interval $(0,\infty)$.

(b) $f$ has a local minimum at $0$.

(c) $\lim_{x\to 0^-}f(x)=\infty$.

(d) $f(x)<0$ for $x<0$.

(e) None of the above statements is true.