Convex Function ⇔ Positive Semidefinite etc.

Let f (x) be a twice differentiable function in n variables defined on an open convex set S. Then we have:

1. f''(x) is positive semidefinite for all x ∈ S ⇔ f is convex in S
2. f''(x) is negative semidefinite for all x ∈ S ⇔ f is concave in S
3. f''(x) is positive definite for all x ∈ S ⇔ f is strictly convex in S
4. f''(x) is negative definite for all x ∈ S ⇔ f is strictly concave in S

http://home.bi.no/a0710194/Teaching/BI-Mathematics/GRA-6035/2010/lecture5-hand.pdf