Is the hessian negative semi-definite if we have an interior maximum? Is it true that given a smooth scalar field f on a domain D , if f attains a maximum (minimum)
on the interior of D then the hessian of f evaluated at this max (min)  is negative (positive) semi-definite?
I have seen this quoted as a fact but usually when i try to find a proof all that comes up is the second derivative test (given hessian is negative definite then we have max)
is this actually just equivalent to the test?
EDIT:
if we consider an approximation via taylor's theorem we have (near $ a $ ) :
$$ f(x) = f(a) + Df(a)^{T} (x-a)  + (x-a)^{T} D^2 f(a) (x-a) + o(|x-a|^3)$$
now if $a$ is max then$ f(x) \le f(a) $ so $ D^2 f(a) $ is negative semidef ?

I've removed a previous answer that didn't answer the question.
Edit: Yes, this follows from the proof of Theorem 9.6(c) on p. 311 of Calculus, Vol. 2, by Apostol. 
