I am currently struggling on the following problem. Let $\sigma>0$ a real number and $g_\sigma$ the gaussian function with mean 0 and deviation $\sigma$. Let also $f$ be a smooth real function such that:
- $f$ is negative before 0
- $f$ is positive after 0
- $f'(0)>0$
We know that it exists a unique $x_\sigma\in\mathbb{R}$ such that $(f*g_\sigma)(x_\sigma)=0$. Can one prove that $(f*g_\sigma)'(x_\sigma)>0$ ?
Thank you very much for the help!