Extremum of a monotonic function combination

Question

If solution of $f'(x) /g'(x)= \lambda $ is an extremum point of $y = f(x) - \lambda \,g(x) $ then can it be shown that $f(x)$ and $g(x)$ are monotone functions?

Answer 1

Not even locally, no. A 'simple' counter-example can be obtained by taking $\lambda=0$ and $f$ given by

$$f(x)=x^2\cdot \exp\left({\sin\left(\frac1x\right)}\right),$$

where of course $f(0)=0$. Then $f(x)>0$ when $x\neq 0$, so that $x=0$ is a (global) minimum, but $f$ is never monotone near $x=0$.

You can check that $f$ is everywhere differentiable, although its derivative is not continuous. That said, if you take $f(x)=x^4\cdot \exp\left({\sin\left(\frac1x\right)}\right)$, then all of these hold and additionally $f'$ is everywhere continuous (differentiable in fact).