The Main Idea 

Finding where a function reaches its highest or lowest values is one of the most practical applications of calculus. For functions of two variables, the process mirrors single-variable calculus but requires checking both partial derivatives.

What Are Critical Points?

A point [latex](x_0, y_0)[/latex] is a critical point of [latex]f(x, y)[/latex] if:

1. Both partial derivatives are zero: [latex]f_x(x_0, y_0) = 0[/latex] and [latex]f_y(x_0, y_0) = 0[/latex], OR
2. Either [latex]f_x(x_0, y_0)[/latex] or [latex]f_y(x_0, y_0)[/latex] doesn’t exist

Critical points are candidates for local maxima and minima.

Local Extrema

• A local maximum at [latex](x_0, y_0)[/latex] means [latex]f(x_0, y_0)[/latex] is greater than or equal to all nearby function values
• A local minimum at [latex](x_0, y_0)[/latex] means [latex]f(x_0, y_0)[/latex] is less than or equal to all nearby function values
• If the inequality holds everywhere in the domain, you have a global (absolute) maximum or minimum

Problem-Solving Strategy

1. Calculate both partial derivatives [latex]f_x[/latex] and [latex]f_y[/latex]
2. Set both equal to zero and solve the system of equations
3. Check where partials don’t exist (look for division by zero, square roots of negatives, etc.)
4. Verify your solutions by substituting back into the original equations

Key Insight from Fermat’s Theorem

If [latex]f[/latex] has a local extremum at [latex](x_0, y_0)[/latex] and both partial derivatives exist there, then [latex](x_0, y_0)[/latex] must be a critical point. This tells you where to look for extrema, but not every critical point is an extremum—you’ll need additional tests to determine the nature of each critical point.

The Main Idea 

Once you’ve found critical points, you need to determine whether each one is a local maximum, local minimum, or saddle point. The second derivative test gives you a systematic way to classify them using second-order partial derivatives.

The Discriminant Formula

At a critical point [latex](x_0, y_0)[/latex], calculate:

[latex]D = f_{xx}(x_0, y_0)f_{yy}(x_0, y_0) - (f_{xy}(x_0, y_0))^2[/latex]

This discriminant [latex]D[/latex] replaces the single second derivative from single-variable calculus.

Interpreting the Test

• If [latex]D > 0[/latex] and [latex]f_{xx} > 0[/latex]: Local minimum (bowl-shaped, curves up)
• If [latex]D > 0[/latex] and [latex]f_{xx} < 0[/latex]: Local maximum (dome-shaped, curves down)
• If [latex]D < 0[/latex]: Saddle point (curves up in one direction, down in another)
• If [latex]D = 0[/latex]: Test is inconclusive (need other methods)

What’s a Saddle Point?

A saddle point is where both partial derivatives are zero but the point is neither a maximum nor minimum. Picture a horse saddle: it curves up in one direction and down in another. The point [latex](0, 0)[/latex] on [latex]f(x, y) = x^2 - y^2[/latex] is a classic example.

Problem-Solving Strategy

1. Find all critical points by solving [latex]f_x = 0[/latex] and [latex]f_y = 0[/latex] simultaneously
2. Calculate all three second partial derivatives: [latex]f_{xx}[/latex], [latex]f_{yy}[/latex], and [latex]f_{xy}[/latex]
3. Evaluate [latex]D[/latex] at each critical point
4. Apply the test to classify each point

Key Reminder

When [latex]D > 0[/latex], check the sign of [latex]f_{xx}[/latex] (or [latex]f_{yy}[/latex]—either works). Positive means minimum, negative means maximum.

The Main Idea 

Finding absolute (global) extrema on a closed, bounded region requires checking both the interior and the boundary of your domain. Unlike local extrema, absolute extrema compare function values across the entire region.

The Extreme Value Theorem

A continuous function on a closed, bounded set is guaranteed to have both an absolute maximum and an absolute minimum somewhere in that region. Your job is to find where they occur.

Where to Look

Absolute extrema can only occur in two places:

• Interior critical points: where both [latex]f_x = 0[/latex] and [latex]f_y = 0[/latex]
• Points on the boundary: the edge of your domain

Problem-Solving Strategy

1. Find interior critical points: Solve [latex]f_x = 0[/latex] and [latex]f_y = 0[/latex] simultaneously, keeping only points inside your domain
2. Evaluate [latex]f[/latex] at each interior critical point
3. Find extrema on the boundary: Parameterize the boundary and find maximum/minimum values there (this is often the hardest step)
4. Compare all values: The largest is your absolute maximum, the smallest is your absolute minimum

Handling the Boundary

For rectangular domains, check each edge separately by treating one variable as constant. For circular boundaries, use parametric equations like [latex]x = r\cos t[/latex], [latex]y = r\sin t[/latex]. Don’t forget to check corner points on rectangular domains—they’re often where extrema occur.

Key Insight

The absolute extrema might not be at critical points. They frequently occur on the boundary, which is why checking the boundary thoroughly is essential. Compare every candidate value you find—no shortcuts.