Minimum of sum of convex functions: single dimension

Question

Let $g_1(x)$ and $g_2(x)$ be convex functions from $\mathbb{R}\rightarrow \mathbb{R}$, and assume that the global minimum of each $g_i$, denoted $x_i^*$, is unique. with $x_1^*<x_2^*$. Does the minimum of $ g_1(x) + g_2(x)$ lie in $[x_1^*,x_2^*]$?

Answer 1

Yes. If $x < x^*_1$ then $g_1(x) > g_1(x^*_1)$ and $g_2(x) > g_2(x^*_1)$ (because $g_2$ is decreasing on $(-\infty, x^*_2)$), so $g_1(x) + g_2(x) > g_1(x^*_1) + g_2(x^*_1)$. Similarly if $x > x^*_2$.