Second Derivative Test for Partial Derivatives

1. Dec 19, 2011

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

2. Dec 20, 2011

HallsofIvy

Staff Emeritus
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.