A ratio of two convex functions with different minima cannot be monotone. Proof?

Question

Let $\lambda(x)=\frac{f(x)}{g(x)}$ where $f(x)$ is a differentiable function minimized at $x=x_1$ and $g(x)$ is a differentiable function minimized at $x=x_2\neq x_1$. How can I show that $\lambda(x)$ is not a monotone function?

Answer 1

Try e.g. $g(x)=x^2+1$, $f(x)=(\pi+\arctan(x)) g(x)$. Then $f(x)/g(x) = \pi+\arctan(x)$ is monotone increasing. $f$ has a minimum near $-0.168$ while $g$ has a minimum at $0$. Moreover, $f$ and $g$ are convex.