Summary of Local Extrema

We begin the summary with a few rigorous definitions.

A local maximum of f(x,y) is a point (x₀,y₀) such that $f(x,y) \le f(x_0,y_0)$ at all points near (x₀,y₀).

A local minimum of f(x,y) is a point (x₀,y₀) such that $f(x,y) \ge f(x_0,y_0)$ at all points near (x₀,y₀).

A saddle point is a point (x₀,y₀) on the function such that within any distance, no matter how small, there exist points (x₁,y₁) and (x₂,y₂) such that $f(x_1,y_1) \ge f(x_0,y_0)$ and $f(x_2,y_2) \le f(x_0,y_0)$. Thus, in some directions, the function is increasing, while in others it is decreasing.

A critical point of the function f(x,y) is any point (x₀,y₀) such that $\mbox{grad}f(x_0,y_0) = \vec{0}$.

To find all local extrema of f(x,y) follow these steps.

1.

Calculate $\mbox{grad}f$.

2.

Find all critical points. In other words, locate all ordered pairs (x,y) such that $\mbox{grad}f = \vec{0}$.

3.

Compute the discriminant, D = f_xxf_yy - f_xy².

4.

Evaluate D at each critical point, and apply the second derivative test outlined above to classify each critical point.