I was reading a paper and the autor did the following inequality, but I couldn't understand why:
If $u:\mathbb{R}^n\to\mathbb{R}$ is smooth and bounded then $$|u(x+y)-u(x)-\nabla u(x)\cdot y|\leq C||D^2u||_{\infty}|y|^2$$
PS: Using the Mean Value Inequality, I could only get that $$|u(x+y)-u(x)-\nabla u(x)\cdot y|\leq C||Du||_{\infty}|y|$$
Any help would be appreciated.