If $x^{-} = \textrm{argmin}_x f(x) $, $x^{+} = \textrm{argmin}_x g(x) $, and $x^{-} \neq x^{+} $, then can we state the following:

Question

If $x^{-} = \operatorname{argmin}_x f(x) $, $x^{+} = \operatorname{argmin}_x g(x) $, and $x^{-} \neq x^{+} $, then can we state that $x^* \neq x^-, x^+$ where $x^{*} = \operatorname{argmin}_x f(x) + g(x)$.

If the optimal point for $f(x)$ and $g(x)$ are different. Then can say that the optimal point for their summation is none of their optimal points, assuming that both functions are non-zero.

If this does not hold, are there conditions under which they hold?

As suggested in the comments, $f$ and $g$ have unique minimizers.

Answer 1

The answer is no. For instance, $f(x)=|x|$ has a minimizer at $0$ and $g(x)=0.5|x-1|$ has a minimizer at $1$; however $f(x)+g(x)$ has a minimizer at $0$.

Look at this graph for a demonstration. Note that both $f$ and $g$ are convex in this example.