Consider the quadratic function
g(x, y) = Ax2 + 2Bxy + Cy2 .
The only critical point is (0, 0) and the second derivatives of g(x, y) are
gxx = 2A
gxy = 2B
gyy = 2C .
The second derivative test states that if at (0, 0) we have gxx < 0 and gxx gyy − g2xy > 0, then (0, 0) is a local minimum. For our function g(x, y) this means
2A < 0 and 4AC − 4B2 > 0 .
When we divive the first inequality by 2 and the second one by 4, we recover the maximum-minimum test for quadratic functions. The same argument can be used for the other types of critical points.
Note: There is one slight difference between the two tests though. The second derivative test tells us when a critical point is a local maximum or minimum. For a quadratic function it is true that a local maximum or minimum is in fact a global one.
