I am struggling to show the following problem. Let $f \in \mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ (satisfying the exponential growth condition) be a real function and $x^*\in \mathbb{R}$ such that $f$ is strictly positive before $x^*$, strictly negative after. Formally,
$$\forall x \in \mathbb{R}, f(x) > 0 \text{ if } x<x^*, f(x) <0\text{ if } x>x^*.$$
More generally, we suppose $f$ changes sign only once.
Can one show that the the function $g$ defined as :
$$g(x) := \int_\mathbb{R}f(x+s)e^{-s^2/2}ds$$
has a unique root on $\mathbb{R}$ ?
I am pretty sure that it is true. Any hints, solutions or even counter-examples would be highly appreciated ! Thank you very much. (I am not familiar with the properties of the convolutions.)
If the statement is false, could you provide some stronger conditions on $f$ that makes it true ?
Examples
If $f(x) = -x$, then $g(x) = -x$ and the result holds. If $f = \arctan$, the figure shows that the claim still holds:
With more complex function (non-increasing) such as :
$$f(x) = \frac{x}{1+x^2}$$
we have:
