Lipschitz convex functions

Question

I am trying to think of a differentiable Lipschitz convex function of $2$ variables other than the linear function, but I am unable to think of one. Can I have some examples, please?

Answer 1

What about $$ f(x) = \sqrt{ \|x\|^2 + \epsilon} $$?

Answer 2

Think about any differentiable function whose derivative is bounded

Like $\sin(x + y)$ or $\cos (x + y)$