Section 7.3: Maxima and Minima

Local Extrema

• f(a, b) is a local maximum if there exists a circular region in the domain of f with (a, b) as the center such that:

  f(a, b) ≥ f(x, y)

  for all (x, y) in the region.
• f(a, b) is a local minimum if there exists a circular region in the domain of f with (a, b) as the center such that:

  f(a, b) ≤ f(x, y)

  for all (x, y) in the region.
• If f(a, b) is either a local maximum or a local minimum, it is called a local extremum.

Local Extrema and Partial Derivatives

Let f(a, b) be a local extremum (a local maximum or a local minimum) for the function f. If both f_x and f_y exist at (a, b), then:

f_x(a, b) = 0andf_y(a, b) = 0

Points (a, b) for which these conditions hold are called critical points.

Second-Derivative Test for Local Extrema

If:

1. z = f(x, y)
2. f_x(a, b) = 0 and f_y(a, b) = 0 [(a, b) is a critical point.]
3. All second-order partial derivatives of f exist in some circular region containing (a, b) as center.
4. A = f_xx(a, b), B = f_xy(a, b), C = f_yy(a, b)

Then:

Case 1. If AC - B² > 0 and A < 0, then f(a, b) is a local maximum.

Case 2. If AC - B² > 0 and A > 0, then f(a, b) is a local minimum.

Case 3. If AC - B² < 0, then f has a saddle point at (a, b).

Case 4. If AC - B² = 0, the test fails.