$f(x)$ and $g(x)$ both have a global minimum, and we take the weighted sum of them like the following:
$$ w \cdot f(x) + (1-w) \cdot g(x)$$
Will this also be convex? And since this is a linear combination, does it suffice to say that the minimum is
$$ w \cdot f_{min} +(1-w) \cdot g_{min}$$
Any suggestion would be appreciated.
If $f$ and $g$ are convex then the sum of convex function is convex, hence yes, it is convex.
However, let $f(x)=2x^2$ and $g(x) = (x-1)^2$
then $\frac12f(x) + \frac12g(x)=x^2+\frac12(x-1)^2$
the minimal pont occurs when $2x+(x-1)=0$
$$x=\frac13$$
and the corresponding value is $\frac19+\frac12\cdot \frac49 >0$