my question is: Can I use Rolle's theorem to prove that the second derivative has a zero?
Consider a Real function of Real variable defined by $f(x)=(x+1)\cdot e^{x^2}$. Prove that the second derivative has a root.
So I did the $f ' (x)$.
$f ' (x) = e^{x^2}(2x^2 + 2x + 1)$
The $f ' (x)$ is continuous on $\mathbb R$.
$f ' (x)$ is differentiable.
$f ''(x) = (4x^3+4x^2+6x+2)\cdot e^{x^2}$
and $f ' (- \infty) = f ' (+\infty)$
So for the Rolle Theorem there is a point $c$ that belongs at $(- \infty, + \infty)$ such that $f ''(c) = 0$
is this correct?