Smallest Lipschitz constant of gradient of a function

Question

I have to show that $2\Vert{A}\Vert$ is the smallest Lipschitz constant of $\nabla f$, where $f(x)=x^{T}Ax+2b^{T}x+c$.

$\forall x,y\in \mathbb{R}^n, \Vert\nabla f(x)-\nabla f(y)\Vert = \Vert2(Ax+b)-2(Ay+b)\Vert = \Vert2A(x-y)\Vert \le \Vert2A\Vert \Vert x-y\Vert$

but I don't know how to show that it's the smallest.

Answer 1

You can take $y = 0$ and then maximize over all $x$ with $||x|| = 1$ to get that $2||A||$ is the best constant.