Calc 3 Lecture NotesSection 12.7Page 1 of 11

Section 12.7: Extrema of Functions of Several Variables

Big idea: Partial derivatives can be used to find the extrema of functions of two variables, just as derivatives could be used to find extrema of functions of a single variable.

Big skill: You should be able to find extrema and saddle points of functions of two variables.

Definition 7.1: Local Extrema

We call f(a, b) a local maximum of f if there is an open disk R cantered at point (a, b) for which f(a, b) f(x, y) for all (x, y) R. Similarly, we call f(a, b) a local minimum of f if there is an open disk R cantered at point (a, b) for which f(a, b) f(x, y) for all (x, y) R. In either case,
f(a, b) is called a local extremum.

Definition 7.2: Critical Point

The point (a, b) is a critical point of the function f(x, y) if (a, b) is in the domain of f and eitheror one or both of and do not exist at (a, b).

Theorem7.1: Condition for a Local Extremum

If f(x, y) has a local extremum at (a, b), then (a, b) must be a critical point of f.

Practice:

  1. Find/investigate the critical points of the function
  1. Find/investigate the critical points of the function

  1. Find/investigate the critical points of the function .

  1. Find/investigate the critical points of the function .
  1. Find/investigate the critical points of the function .
  1. Find/investigate the critical points of the function .

Definition 7.3: Saddle Point

The point P(a, b, f(a, b)) is a saddle point of z = f(x, y) if (a, b) is a critical point of f and if every open disk centered at (a, b) contains points (x, y) in the domain of f for which f(x, y) < f(a, b) and points (x, y) in the domain of f for which f(x, y) > f(a, b).

Theorem7.2: Second Derivatives Test

Suppose that f(x, y) has continuous second-order partial derivatives in some open disk containing the point (a, b) and that fx(a, b) = fy(a, b) = 0. Define the discriminant D for the point (a, b) by:

D(a, b) = fxx(a, b)fyy(a, b) – [fxy(a, b)]2.

(i).If D(a, b) > 0 and fxx(a, b) > 0, then f has a local minimum at (a, b).

(ii).If D(a, b) > 0 and fxx(a, b) < 0, then f has a local maximum at (a, b).

(iii).If D(a, b) < 0, then f has a saddle point at (a, b).

(iv).If D(a, b) = 0, then no conclusion can be drawn.

Practice:

  1. Find and classify the critical points of the function

Practice:

  1. Find the shortest distance from the point to the plane .`

  1. A rectangular box with no lid is to be constructed from 12 square meters of cardboard by cutting square notches out of each of the four corners then folding up the remaining sides. Find the maximum volume that such a box can have.

One very useful application of max/min theory for functions of two variables is the statistical technique of linear regression, which is a particular case of a more general technique called least squares. Linear regression involves finding a line of “best fit” for a given set of data points. The goal is to find the line that minimizes the sum of the squares of the vertical distances between the data points and the line. This problem can be formulated as a minimization problem involving finding the absolute minimum of a certain function f (the square of the residuals) of the two variables m and b.

You can show that, in general, for n data points (x1, y1), (x2, y2), …, (xn, yn), the linear least square fit yields the two equations

Which have solution

Practice:

  1. Find the linear least squares fit for the data (1, 5), (2, 11), (3, 14), (4, 22), (5, 22).

Method of Steepest Ascent to find a Local Extrema of z = f(x, y):

  1. Pick a starting guess.
  2. “Move away” from in the direction of until you find new coordinates such that .
  3. Repeat step 2 until you are close enough.

Practice:

  1. A man stands at the point on a hill whose elevation is given by . Use the method of steepest ascent to find an approximation of the shortest path to the top, and trace out the path on the level curves shown below.

Definition 7.4: Absolute Extrema

We call f(a, b) the absolute maximum of f if f(a, b) f(x, y) for all (x, y)  domain. Similarly, we call f(a, b) the absolute minimum of f if f(a, b) f(x, y) for all (x, y)  domain.

Theorem 7.3: Extreme Value Theorem

Suppose that f(x, y) is continuous on a closed and bounded region . Thenf has both an absolute maximum and absolute minimum on R.

Practice:

  1. Find/investigate the critical points of the function on the region R = {(x, y) | -2  x  2, 0  y  4}