$E$ is a normed vector space and $f:E \rightarrow \mathbb{R} \in C^{2}$

Question

Let $E$ be a normed vector space and $f:E \to \mathbb{R}$ a $C^2$-function such that $\forall x \in E,~ f(x)>0$. Suppose that $\exists M>0$ such that $\forall x \in E, \|D^2f(x)\|\leq M$.

1. prove that if $h \in E$ and $\lambda \in \mathbb{R}$, we have $$f(x)+\lambda Df(x)h + \frac{\lambda^{2}}{2}M||h||^2 >0, \quad \forall x \in E$$

2. Deduce that $ \|Df(x)\| \leq \sqrt{2Mf(x)}$

Answer 1

By Taylor's theorem: $$\begin{align*} 0 &\leq f(x+ \lambda h) = f(x)+f'(x) \cdot \lambda h + \dfrac{1}{2} \int_0^1 dt\ f''((1 - t)x + th) \cdot (\lambda h, \lambda h) \\ &\leq f(x) + \lambda f'(x) \cdot h + \lambda^2 \dfrac{M}{2} \|h\|^2 \end{align*}$$