Problem 21 - Calculus an Applied Approach

Question

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function \(f(x, y)\) at the critical point \(\left(x_{0}, y_{0}\right) .\) $$ f_{x x}\left(x_{0}, y_{0}\right)=9, f_{y y}\left(x_{0}, y_{0}\right)=4, f_{x y}\left(x_{0}, y_{0}\right)=6 $$

Step-by-Step Solution

Answer

There is insufficient information to determine whether \(f(x, y)\) has a relative maximum, minimum, or a saddle point at the given critical point \(\left(x_{0}, y_{0}\right).\)

Step 1: Compute the Discriminant

First, compute the discriminant, denoted as D, using the formula: \(D = f_{xx}(x_{0}, y_{0}).f_{yy}(x_{0}, y_{0}) - [f_{xy}(x_{0}, y_{0})]^2\). Plugging the given values into this formula, this results in: \(D = 9 * 4 - 6^2\).

Step 2: Carry out the Calculations on Discriminant

Next, calculate the discriminant: \(D = 36 - 36 = 0\).

Step 3: Analyze the Discriminant and Second Partial Derivatives

To make a decision about the nature of the critical point, three conditions need to checked using the sign of the discriminant and the sign of the second partial derivatives. If D > 0 and \(f_{xx}(x_{0}, y_{0}) > 0\), the point is a relative minimum. If D > 0 and \(f_{xx}(x_{0}, y_{0}) < 0\), the point is a relative maximum. If D < 0, the point is a saddle point. If D = 0, there's insufficient information to determine the nature of the critical point. Since in this case, D is zero, no conclusion can be drawn about whether \((x_{0}, y_{0})\) is a relative maximum, minimum, or saddle point.

Key Concepts

Second Partial Derivative Test

The Second Partial Derivative Test is a mathematical tool used to determine the nature of critical points in functions of two variables. This test uses second derivatives, which are derivatives of the derivative, to evaluate the curvature of the function at a given point.

To apply the test, you calculate the partial derivatives of the function:

• First, find the second partial derivatives, \(f_{xx}, f_{yy},\) and \(f_{xy}\).
• Then, use these to compute the discriminant, \(D\), with the formula: \[D = f_{xx}(x_{0}, y_{0}) \cdot f_{yy}(x_{0}, y_{0}) - [f_{xy}(x_{0}, y_{0})]^2\].

Depending on the value of \(D\):

• If \(D > 0\) and \(f_{xx} > 0\), there's a relative minimum.
• If \(D > 0\) and \(f_{xx} < 0\), there's a relative maximum.
• If \(D < 0\), it's a saddle point.
• If \(D = 0\), the test is inconclusive.

Understanding and using the Second Partial Derivative Test allows you to analyze and classify critical points effectively.

Discriminant in Multivariable Calculus

In multivariable calculus, the discriminant helps to determine the nature of relative extrema at critical points. It's a pivotal quantity derived from the second partial derivatives of a function with two variables.

For a function \(f(x, y)\), the discriminant \(D\) is given by: \[D = f_{xx}(x_{0}, y_{0}) \cdot f_{yy}(x_{0}, y_{0}) - [f_{xy}(x_{0}, y_{0})]^2\].

Here’s what each term represents:

• \(f_{xx}\) is the second derivative of \(f\) with respect to \(x\), measuring how the function curves in the \(x\)-direction.
• \(f_{yy}\) is the second derivative with respect to \(y\), indicating the curvature in the \(y\)-direction.
• \(f_{xy}\) is the mixed derivative, capturing how changing \(x\) affects \(f\) when \(y\) is also changing.

This combination of derivatives in \(D\) allows us to assess if a point is a max, min, saddle point, or if more information is needed.

Saddle Point Analysis

A saddle point is a type of critical point that isn't a local maximum or minimum. Understanding saddle points is crucial because they indicate points where the behavior of the function changes from increasing to decreasing or vice versa.

Saddle points occur when the discriminant \(D < 0\) in the Second Partial Derivative Test. This suggests that while the function curves up in one direction, it simultaneously curves down in another.

Key characteristics of saddle points include:

• They resemble a horse's saddle, curving upwards in one direction and downwards in the perpendicular direction.
• No matter how the point is approached, it never turns into a peak or a trough in all directions.
• Their presence indicates that there are no clear relative extrema; the function has no definitive max or min at that point.

Although saddle points don't mark traditional peaks or troughs, understanding their nature helps in the contour and surface analysis of multivariable functions.