Professor Leonard – Calculus 3
13.8 – Finding Extrema of Functions of 2 Variables (Max and Min)
Thomas – 14.7 Extrema and Saddle Points
Quick Review of Extrema
In Calculus 1, we learned the local (relative) extrema are points that are higher (local maxima) or lower (local minima) than all nearby points on the curve. Absolute extrema are the highest and lowest values attained by a function over its entire domain. For one-variable functions, we think of peaks and valleys along a curve. For functions of two variables, we simply extend these same concepts to surfaces. Unlike one-variable functions, surfaces can also contain saddle points – points that are neither maxim nor minima.
Finding Critical Points
Local extrema and saddle points can only occur at critical points. For differentiable functions, critical points occur where both first partial derivatives equal zero: f_x(x,y)=0,\qquad f_y(x,y)=0. Solving these two equations gives every critical point of the surface. At this stage, we cannot determine whether a critical point is a maximum, minimum, or saddle point. To classify each critical point, we perform the Second Derivative Test.
The Second Derivative Test
First partial derivatives tell us about slope and where the surface becomes flat. Second derivatives describe concavity, or how the surface bends around those flat points.
We first calculate D=f_{xx}f_{yy}-(f_{xy})^2. This quantity is known as the Hessian determinant. We then evaluate D at each critical point.
If D>0\text{ and }f_{xx}>0:
- The surface curves upward in every direction.
- A local minimum exists.
If D>0\text{ and }f_{xx}<0:
- The surface curves downward in every direction.
- A local maximum exists.
If D<0:
- The surface curves upward in some directions and downward in others.
- A saddle point exists.
If D=0:
- The Second Derivative Test is inconclusive.
- Another method is required to classify the critical point.
Simplified Steps for Finding Local Extrema
- Find the first partial derivatives:
f_x,\qquad f_y - Solve:
f_x=0,\qquad f_y=0
to find all critical points. - Compute the second partial derivatives:
f_{xx},\qquad f_{yy},\qquad f_{xy} - Calculate:
D=f_{xx}f_{yy}-(f_{xy})^2 - Evaluate D at each critical point and classify it as a local maximum, local minimum, or saddle point.
- If requested, evaluate:
f(x,y)
at each local maximum and minimum to determine their corresponding function values.
Absolute Maximum and Minimum
Local extrema only describe what happens near a point. To find absolute maximum and minimum, we need to consider the entire domain of a function. For any continuous function defined on a closed and bounded region, both an absolute maximum and an absolute minimum must exist. These extrema may occur:
- at an interior critical point,
- somewhere along a boundary,
- or at a corner (endpoint) of the region.
A function may have more than one point where its absolute maximum or minimum value occurs.
Finding Absolute Maxima and Minima
Begin by sketching the domain whenever possible. Find all the interior critical points and evaluate the function at each.
Next, examine each boundary separately. Along each boundary, one variable remains constant while the other varies. As such, we can reduce the problem to a single-variable function and locate any critical points along that boundary.
Doing our work properly, we have evaluated the function at
- each boundary critical point,
- each endpoint (corner),
- and every interior critical point.
We can then compare every function value and identify the values that are the absolute maximum and minimum.