Maxima and Minima Problems in Multivariable Calculus

This guide covers the identification and classification of extrema for functions of two variables, extending the concepts of single-variable calculus to surfaces in 3D space.

Learning Objectives

• Locate Critical Points: Identify points where the gradient is zero or undefined.
• Classify Local Extrema: Apply the Second Derivative Test to distinguish between local maxima, local minima, and saddle points.
• Find Absolute Extrema: Determine the global maximum and minimum values on a closed and bounded domain by checking critical points and boundary curves.

Key Terms & Glossary

• Critical Point: A point $(a, b)$ in the domain of $f(x, y)$ such that $f_x(a, b) = 0$ and $f_y(a, b) = 0$, or at least one of these partial derivatives does not exist.
• Local Maximum: A point $(a, b)$ where $f(a, b) \geq f(x, y)$ for all $(x, y)$ in a small disk centered at $(a, b)$.
• Saddle Point: A critical point $(a, b)$ that is neither a local maximum nor a local minimum; the surface curves up in one direction and down in another.
• Discriminant ($D$): A value calculated from second-order partial derivatives used to classify critical points.

The "Big Idea"

In single-variable calculus, we look for "peaks" and "valleys" along a line. In multivariable calculus, we look for these features on a surface. Because we have more directions to move, a point can be a "peak" in the $x$-direction but a "valley" in the $y$-direction—this unique multivariable phenomenon is called a Saddle Point. Optimization allows us to find the most efficient configurations in physics, economics, and engineering.

Formula / Concept Box

Hierarchical Outline

• I. Identification of Critical Points
  • Partial Derivatives: Calculate first-order derivatives $f_x$ and $f_y$.
  • Systems of Equations: Solve $f_x=0$ and $f_y=0$ simultaneously to find coordinates $(x, y)$.
• II. Classification using the Second Derivative Test
  • Hessian Components: Calculate $f_{xx}$, $f_{yy}$, and $f_{xy}$.
  • Discriminant Analysis: Evaluate $D$ at each critical point.
• III. Absolute Extrema on Closed Domains
  • Interior Check: Test all critical points inside the region.
  • Boundary Check: Parameterize the boundary (e.g., $y=g(x)$) and treat as a 1D optimization problem.
  • Comparison: Compare values at critical points, boundary extrema, and corner points.

Visual Anchors

Optimization Process Flow

Geometric Interpretation of a Saddle Point

Definition-Example Pairs

• Term: Second Derivative Test
• Definition: A method to determine the nature of a critical point by looking at the surface's curvature in multiple directions.
• Example: For $f(x, y) = x^2 - y^2$, $f_x=2x, f_y=-2y$. The point $(0,0)$ is a critical point. $f_{xx}=2, f_{yy}=-2, f_{xy}=0$. $D = (2)(-2) - 0^2 = -4$. Since $D < 0$, it is a Saddle Point (like a mountain pass).

Worked Examples

Finding Local Extrema

Problem: Find and classify the critical points of $f(x, y) = x^3 - 3x + y^2$.

Step 1: Find Critical Points

• $f_x = 3x^2 - 3 = 0 \implies x^2 = 1 \implies x = 1, -1$
• $f_y = 2y = 0 \implies y = 0$
• Critical Points: $(1, 0)$ and $(-1, 0)$

Step 2: Calculate Second Derivatives

• $f_{xx} = 6x$
• $f_{yy} = 2$
• $f_{xy} = 0$

Step 3: Apply the Test

• At (1, 0): $D = (6)(2) - 0^2 = 12$. Since $D > 0$ and $f_{xx} = 6 > 0$, $(1, 0)$ is a Local Minimum.
• At (-1, 0): $D = (-6)(2) - 0^2 = -12$. Since $D < 0$, $(-1, 0)$ is a Saddle Point.

Checkpoint Questions

1. If $D(a, b) > 0$ and $f_{xx}(a, b) < 0$, what type of extremum is present at $(a, b)$?
2. True or False: A saddle point occurs when the function has a local maximum in one direction and a local minimum in another.
3. What must you check in addition to critical points when finding the absolute extrema on a closed triangular region?

[!TIP] When solving $f_x=0$ and $f_y=0$, always check if your solutions for $x$ depend on $y$ and vice-versa. Don't forget the corners of your domain when checking boundaries!