Question

Use the Second Derivative Test to prove that if \((a, b)\) is a critical point of \(f\) at which \(f_{x}(a, b)=f_{y}(a, b)=0\) and \(f_{x x}(a, b)<0

Short answer

Question: Prove that if \(f_{x x}(a, b)<0<f_{y y}(a, b)\) or \(f_{y y}(a, b)<0<f_{x x}(a, b),\) then the function \(f\) has a saddle point at \((a,b)\). Answer: Both conditions lead to a negative determinant of the Hessian matrix, \(D(a,b)<0\), which indicates a saddle point at \((a,b)\) according to the Second Derivative Test.

Step-by-step solution

Recall the Second Derivative Test conditions

In order to apply the Second Derivative Test, we need to find the determinant of the Hessian matrix evaluated at the point \((a,b)\). The Hessian matrix of a function \(f\) is given by: $\begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix}$ The determinant of the Hessian matrix is given by: \(D(a,b) = (f_{xx}(a,b))(f_{yy}(a,b)) - (f_{xy}(a,b))(f_{yx}(a,b))\)

Derive the conditions for a saddle point

If \(D(a,b)<0\), then the critical point is a saddle point; if \(D(a,b)>0\), then it can be a local minimum or a local maximum (depending on the sign of \(f_{xx}(a,b)\) or \(f_{yy}(a,b)\)); if \(D(a,b)=0\), then the test is inconclusive. Given the conditions \(f_{x x}(a, b)<0<f_{y y}(a, b)\) or \(f_{y y}(a, b)<0<f_{x x}(a, b),\) we can analyze both cases to see if they lead to a saddle point.

Analyze the first condition

Consider the first condition: \(f_{x x}(a, b)<0 0\), their product is negative. Furthermore, we note that \((f_{xy}(a,b))(f_{yx}(a,b))\) is always nonnegative, since the square of any real number is nonnegative. Thus, \(D(a,b) = \text{negative} - \text{nonnegative} < 0\) which indicates a saddle point.

Analyze the second condition

Now consider the second condition: \(f_{y y}(a, b)<0 0\) and \(f_{yy}(a,b) < 0\), their product is negative. As in the first case, we note that \((f_{xy}(a,b))(f_{yx}(a,b))\) is always nonnegative. Thus, \(D(a,b) = \text{negative} - \text{nonnegative} < 0\) which also indicates a saddle point. Based on the analysis of both conditions, we can conclude that if \(f_{x x}(a, b)<0<f_{y y}(a, b)\) or \(f_{y y}(a, b)<0<f_{x x}(a, b),\) then the function \(f\) has a saddle point at \((a,b)\).

Key concepts

Hessian matrix

Understanding the Hessian matrix is key to applying the Second Derivative Test. The Hessian matrix for a function of two variables \( f(x, y) \) consists of second-order partial derivatives. It looks like this:

• \( f_{xx} \) – the second partial derivative of \( f \) with respect to \( x \) twice.
• \( f_{yy} \) – the second partial derivative of \( f \) with respect to \( y \) twice.
• \( f_{xy} \) and \( f_{yx} \) – the mixed partial derivatives.

So, the Hessian matrix is:\[\begin{bmatrix}f_{xx} & f_{xy} \f_{yx} & f_{yy}\end{bmatrix}\]Remember, this matrix helps you understand how a function behaves around critical points. It does so by capturing the curvature in both directions. That’s why it’s so crucial for determining whether a point is a local extremum or a saddle point.

Saddle point

A saddle point offers an interesting feature in the world of calculus. It's a point on the graph where the function doesn’t have a local maximum or minimum. Instead, the graph curves upwards in one direction and downwards in the other.
The unique characteristic of a saddle point is related to the changing signs in the second derivatives. In the Second Derivative Test, if the determinant of the Hessian matrix \((D(a, b))\) is negative, it indicates a saddle point. This means the function behaves differently in each of the coordinate directions, providing no extreme values at that point.

Critical point

A critical point in multivariable calculus occurs where the first partial derivatives of a function equal zero. In other words, a point \((a, b)\) is critical if:

• \( f_x(a, b) = 0 \)
• \( f_y(a, b) = 0 \)

Critical points are vital in determining where to look for extrema or stationary points of a function. They can represent peaks, troughs, or even a flat region like a saddle. Further testing, like the Second Derivative Test, helps determine the exact nature of these points.

Determinant of the Hessian

The determinant of the Hessian is a specific value calculated from the Hessian matrix and plays a central role in determining the nature of critical points. The formula uses the components:\[D(a, b) = f_{xx}(a, b) \cdot f_{yy}(a, b) - \left( f_{xy}(a, b) \right)^2\]The result gives insights:

• If \( D(a, b) > 0 \), the point might be a local minimum or maximum, depending on the signs of other second derivatives.
• If \( D(a, b) < 0 \), it indicates a saddle point.
• If \( D(a, b) = 0 \), the test is inconclusive.

This determinant helps predict how the function behaves at and around the critical point, making it a calculation of great importance in calculus.

Partial derivatives

Partial derivatives help break down the behavior of functions with multiple variables by looking at how the function changes with respect to one variable while keeping others constant.
For a function \( f(x,y) \), the partial derivatives \( f_x \) and \( f_y \) reveal how \( f \) changes as \( x \) or \( y \) varies. The second partial derivatives \( f_{xx} \), \( f_{yy} \), and the mixed \( f_{xy} \), and \( f_{yx} \) give further depth into how the function's rate of change itself changes, providing essential information about curvature and stability around certain points on the graph.
These derivatives form the backbone of the Hessian matrix and, consequently, are integral in tests like the Second Derivative Test to determine critical point nature.