properties of convex function

Question

Let $f \colon \mathbb{R}^n \longrightarrow \mathbb{R}$ be a convex function.

Is it true that for every $c \in \mathbb{R}^+$ and for every $x,y \in \mathbb{R}^n$ that $$ f(x+c\cdot y) \leq c \cdot f(x+y) \ ?$$

Please advise and thanks in advance.

Answer 1

This is not true. Let $f(x)=x$, $x=1,y=1, c=\frac12$.

$$1+\frac12 (1) > \frac12 (1+1)$$

Answer 2

Hint: See what happens if $f$ is linear and $f(x)=f(y)=1$.