Question

A function \(f\) has continuous second partial derivatives on an open region containing the critical point \((a, b)\). If \(f_{x x}(a, b)\) and \(f_{y y}(a, b)\) have opposite signs, what is implied? Explain.

Short answer

If \(f_{x x}(a, b)\) and \(f_{y y}(a, b)\) have opposite signs, the critical point \((a, b)\) is a saddle point according to the Second Partial Derivative Test.

Step-by-step solution

Understanding the Second-Partial Derivative (Discriminant) Test

The Second-Partial Derivative Test, also known as the Discriminant Rule, is a method used for determining whether a critical point of a function of two variables is a local maximum, local minimum, or a saddle point. The discriminant value \(D\) is calculated as \(D = f_{x x}(a, b) f_{y y}(a, b) - (f_{x y}(a, b))^2\). If \(D > 0\) and both \(f_{x x}(a, b)\) and \(f_{y y}(a, b)\) are positive, the critical point \((a, b)\) is a local minimum. If \(D > 0\) and both \(f_{x x}(a, b)\) and \(f_{y y}(a, b)\) are negative, the critical point \((a, b)\) is a local maximum. However, if D is less than zero, the critical point \((a, b)\) is a saddle point.

Analyzing the Scenario of Opposite Signs of Second Partial Derivatives

Given that the second partial derivatives \(f_{x x}(a, b)\) and \(f_{y y}(a, b)\) have opposite signs, it means one of them is positive while the other is negative. Substituting these into the formula of discriminant, we can immediately conclude that the discriminant value \(D\) is negative.

Drawing Conclusion based on the Discriminant Value

In the Second-Partial Derivative Test, a negative discriminant value \(D\) implies that the critical point \((a, b)\) is neither a local maximum nor a local minimum, but a saddle point. A saddle point is a point on the surface of the graph of the function where the slopes (derivatives) in orthogonal directions are zero (making it a critical point), but the concavities in those directions are opposite. In simpler terms, the surface curve behaves like a saddle around such points, elevating in one direction while dropping in the other.