Must the result be a convex function when a linear function multiple a convex function?

Question

If $f(x)$ is a linear function,and $g(x)$ is a convex function,and $y(x)=f(x) \times g(x)$,must $y(x)$ still a convex function?assume $f(x)>0$, how to prove it?

Answer 1

Yes, if $g(x)$ is a convex function then $g(x)$ is a convex function.

I think you meant to ask if $y(x)$ is a convex function. If so, the answer is no, simply let $f(x)$ be the constant function equal to $-1$, and let $g(x)$ be any non-linear convex function, e.g. $g(x)=x^2$ on $\mathbb R$.