exists r > 0 such that ∇2f (x) > 0 for any x ∈ B(x0,r).

Question

Let f : U ⊆ Rn → R be a twice continuously differentiable function. Let x0 be an interior point of U such that ∇^2f(x0)> 0(hessian). Prove that there exists r > 0 such that ∇2f (x) > 0 for any x ∈ B(x0,r).

i tried to prove it with minor criteria .

Answer 1

For any continuous function $g$ on a topological space $\{x: g(x) >0\}$ is open. When the domain is a metric space $g(x_0) >0$ implies $B(x_0,r) \subset \{x: g(x) >0\}$ for some $r>0$.