In my recent researches, I encountered functions $f$ satisfying the following functional inequality:

$$ (*)\; f(x)\geq f(y)(1+x-y) \; ; \; x,y\in \mathbb{R}. $$

Since $f$ is convex (because $\displaystyle f(x)=\sup_y [f(y)+f(y)(x-y)]$), it is left and right differentiable. Also, it is obvious that all functions of the form $f(t)=ce^t$ with $c\geq 0$ satisfy $(*)$. Now, my questions:

(1) Is $f$ everywhere differentiable?

(2) Are there any other solutions for $(*)$?

(3) Is this functional inequality well-known (any references (paper, book, website, etc.) for such functional inequalities)?

Thanks in advance