TITLE: Why is $x\log(x)$ convex?
QUESTION [6 upvotes]: Why is $x\log(x)$ convex?

According to the definition it must hold:
$(tx+(1-t)y)\log(tx+(1-t)y)\le tx\log(x)+(1-t)y\log(y)$
for all positive $x,y$ and $t\in[0,1]$
edit: It is allowed to derive, but i have to prove using the definition.

REPLY [10 votes]: Since $f(x)=x \log x$ is $C^2$, you can derive twice, and see that 
$$f''(x)=\frac{1}{x}>0. $$

REPLY [10 votes]: Given your name, you should know that a function is convex if its second derivative is positive...