From lecture 27: Positive Definite Matrices and Minima In one of the examples,
A = 2 6
6 20
So, x'Ax = 2x^2 + 12xy + 20y^2

Supposedly, there is a minima when
2x^2 + 12xy + 20y^2 0
and this is supposed to be equivalent to A being positive definite.

However, when I take the 2nd derivative matrix of the function 2x^2 + 12xy + 20y^2, I get
fxx fxy
fyx fyy
=
4 12
12 40
instead of
2 6
6 20

Why is this so?

Question

From lecture 27: Positive Definite Matrices and Minima In one of the examples, 
A = 2 6 
       6 20 
So, x'Ax = 2x^2 + 12xy + 20y^2 

Supposedly, there is a minima when 
2x^2 + 12xy + 20y^2 > 0 
and this is supposed to be equivalent to A being positive definite. 

However, when I take the 2nd derivative matrix of the function 2x^2 + 12xy + 20y^2, I get 
fxx fxy 
fyx fyy
=
4 12 
12 40 
instead of 
2 6 
6 20 

Why is this so?