Section 4.7 - Maximum and Minimum Values for Two-Variable Functions

Section 4.7 - Maximum and Minimum Values for Two-Variable Functions

Introduction to Extrema

In calculus, the study of maximum and minimum values (collectively called extrema) is essential for understanding the behavior of functions. For functions of two variables, these concepts extend naturally from single-variable calculus, but require additional tools such as partial derivatives and the Hessian matrix.

• Local Maximum: A function f(x, y) has a local maximum at point (a, b) if f(a, b) is greater than nearby values.

• Local Minimum: A function f(x, y) has a local minimum at point (a, b) if f(a, b) is less than nearby values.

• Absolute Maximum/Minimum: The largest/smallest value of f(x, y) over its entire domain.

Critical Points

Critical points are candidates for local extrema. For a function f(x, y), a point (a, b) is a critical point if the first-order partial derivatives both vanish or do not exist:

• $f_x(a, b) = 0$

• $f_y(a, b) = 0$

At a critical point, the function may have a local maximum, minimum, or neither.

Theorem: First Derivative Test for Extrema

If f has a local maximum or minimum at (a, b), and the first-order partial derivatives exist there, then:

• $f_x(a, b) = 0$

• $f_y(a, b) = 0$

Thus, all local extrema occur at critical points or on the boundary of the domain.

Second Derivative Test (Hessian Matrix)

To classify critical points, we use the second derivative test involving the Hessian matrix:

• Let $H = \begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix}$

• Let $D = D(a, b) = f_{xx}(a, b) f_{yy}(a, b) - [f_{xy}(a, b)]^2$

Condition	Classification
$D > 0$ and $f_{xx}(a, b) > 0$	Local minimum
$D > 0$ and $f_{xx}(a, b) < 0$	Local maximum
$D < 0$	Saddle point (neither maximum nor minimum)
$D = 0$	Test is inconclusive

Example: For $f(x, y) = y^2 - x^2$, the critical point at (0, 0) is a saddle point.

Extreme Value Theorem for Functions of Two Variables

If f is continuous on a closed, bounded set $D \subset \mathbb{R}^2$, then f attains an absolute maximum and minimum value at some points in $D$.

• Closed set: Contains all its boundary points.

• Bounded set: Fits within some disk in $\mathbb{R}^2$.

To find absolute extrema of f on a closed, bounded set D:

1. Find values of f at critical points inside D.

2. Find values of f on the boundary of D.

3. The largest value is the absolute maximum; the smallest is the absolute minimum.

Example: Find the absolute maximum and minimum of $f(x, y) = x^2 - 2xy + 2y$ on the rectangle $D = \{(x, y) | 0 \leq x \leq 3, 0 \leq y \leq 2\}$.

Summary Table: Classification of Critical Points

Test Condition	Result
$D > 0$, $f_{xx} > 0$	Local minimum
$D > 0$, $f_{xx} < 0$	Local maximum
$D < 0$	Saddle point
$D = 0$	Inconclusive

Additional info: The notes also reference the Closed Interval Method for finding absolute extrema, which is an extension of the single-variable method to functions of two variables.