Second Derivative Test for Partial Derivatives

[SeannyBoi71]

Hi there, just wanted to make a clarification before my final exam.

The second derivative test for partial derivatives (or at least part of it) states

if D = ∂²f/∂x² * ∂²f/∂y² - (∂²f/∂x∂y)² and (a,b) is a critical point of f, then

a) if D(a,b) > 0 and ∂²f/∂x² < 0, then there is a local max at (a,b)

b)if D(a,b) > 0 and ∂²f/∂x² > 0, then there is a local min at (a,b)

and the other two parts are irrelevant for my question. My question is, do I have to specifically check that ∂²f/∂x² is positive or negative, or can I check that ∂²f/∂y² is positive or negative instead? i.e. does it really matter which one I check? Thank you in advance

 

[HallsofIvy]

Clearly, $\left(\partial^2 f/\partial x\partial y\right)^2$ is positive and you are subtracting it from $\left(\partial^2f/\partial x^2\right)\left(\partial^2 f/\partial y^2\right)$. In order that the difference be positive, $\left(\partial^2f/\partial x^2\right)\left(\partial^2 f/\partial y^2\right)$ must be positive which means that $\partial^2f/\partial x^2$ and $\partial^2 f/\partial y^2$ must have the same sign.