Find the domain of this function through analytical ways

Question

Consider $f(x) = \ln(2 + xe^{x^2})$ in the domain $(a, 0]$ where of course $a < 0$.

I was wondering if it's possible to find $a$ via analytical methods, using theorems and definition of analysis in $\mathbb{R}$. No approximation methods. If not, I will be satisfied with some estimation.

Basically I have to study $2 + x e^{x^2} > 0$ which is not solvable in the usual ways. I thought about using the derivative, but the derivative tells me $e^{x^2}(1 + 2x^2)$ which is never zero. This could tell me when the function is increasing though, that is... always. And it doesn't tell me from where it starts.

Do you know some smart theorems or ways to attack this problem in the less numerical way possible?

Thank you!

Answer 1

Obviously, we must have $a<0$.

Now, as $2+ae^{a^2}=0$, we have $$ae^{a^2}=-2\implies a^2e^{2a^2}=4\iff 2a^2e^{2a^2}=8 $$

Thus, $2a^2=W(8)$ or $a=-\sqrt{\dfrac{W(8)}{2}}\approx -0.896$, where $W$ represents Lambert's W function.

Answer 2

$$g(x)=2+x\exp(x^2)$$ $g $ is increasing, you can apply intermediate value theorem to show that $g(a)=0$ has a unique solution in $\mathbb R$, hence $f$ is defined in the domaine $]a;+\infty[$.