Consider some suffiently often continuosly differentiable function $$f:\mathbb{R} \to \mathbb{R}$$ such that $f''(x) > 0$ everywhere. Then assume $f$ has a global minimum, which we will denote $x_f^*$. Now consider the function $$ g: \mathbb{R} \to \mathbb{R}$$ $$ x \mapsto f(x) + \ln f''(x).$$
If $g$ has a local/global minimum, let us denote it by $x_g^*$. Now I wonder how one could quantify how close $x_g^*$ and $x_f^*$ necessarily would have to be.
For example, let $f(x) = x^2 + u(x)$. Then one gets $g(x) = x^2 + u + \ln (2+u'')$. Taking derivatives, one would then want that
$$ x_f^* = (2x + u')^{-1}(\{0\}) \approx x_g^* \in (2x + u' + \frac{u'''}{2+u''})^{-1}(\{0\})$$
In the case that $u$ is a polynomial of only even degrees, one should get that $\frac{u'''}{2 + u''}$ is small in comparison to $x$, so in this case I guess one would have some closeness of the minima. How can one generalize to arbitrary $u$?