Equation related to Smoothness If you have a differentiable function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ that is $\beta$-smooth (for all $v$ and all $w$, you have $\|\nabla f(v)-\nabla f(w)\| \leq \beta \|v-w\|$), how can you show the equation below?

$f(v) \leq f(w) + \langle\nabla f(w), v-w \rangle + \frac{\beta}{2}\|v-w\|^2$
Have you tried using the mean value theorem for integrals? You can relate $f(v)-f(w)$ to $\nabla f$ and use some simple estimates to get something like the above.