Difference of two functions with a flex at the same point could give rise to a peak?

Question

I have two function $f,g\in C^{k}(I,\mathbb{R})$ with $I\subset\mathbb{R}$. Moreover, I know that they have an inflection point at the same point $x_{I}\in\mathbb{R}$. Could the function $h:=f-g$ admit a peak in $x_i$? In other words, by varying the order $k$ of their differentiability class, can I produce an effect like that?