Worksheet 2.7: Critical Points, Local Extrema, and the Second Derivative Test

Transcription

1 Boise State Math 275 (Ultman) Worksheet 2.7: Critical Points, Local Extrema, and the Second Derivative Test From the Toolbox (what you need from previous classes) Algebra: Solving systems of two equations in two variables. Calc I: Critical points, local extrema (maxima and minima) and the second derivative test for functions of a single variable (from Calc I). Calc III: Computing and evaluating first and second partial derivatives and gradients of functions of two variables. Familiarity with level curves and contour maps. Goals In this worksheet, you will: Find critical points of functions of two variables. Use contour maps to determine whether a point is a local maximum, local minimum, or saddle point. Use the second derivative test to determine whether critical points are local minima, local maxima, or saddle points.

2 Boise State Math 275 (Ultman) Worksheet 2.7: Local Extrema, Critical Points, 2 nd Derivative Test1 Definitions Suppose f (x, y) is a function of two variables, and (a, b) is a point in the domain of f. Local Maximum, Local Minimum (Local Extrema): f (a, b) is a local maximum if: f (a, b) f (x, y) for all points (x, y) near the point (a, b). f (a, b) is a local minimum if: f (a, b) f (x, y) for all points (x, y) near the point (a, b). Saddle Point: (a, b) is a saddle point if: f (a, b) = 0 f (a, b) is neither a local maximum nor a local minimum. (No matter how close you get to (a, b), there are some points (x, y) such that f (a, b) > f (x, y), and others such that f (a, b) < f (x, y).) A saddle point is a two-dimensional version of an inflection point. Some additional terminology: Maxima is the plural form of maximum. Minima is the plural form of minimum. Extremum means extreme value that is, either a maximum or a minimum. The plural form is extrema.

3 Boise State Math 275 (Ultman) Worksheet 2.7: Local Extrema, Critical Points, 2nd Derivative Test2 Model 1: Using Graphs and Contour Maps to Find Local Maxima, Local Minima, and Saddle Points Example 1A: f (x, y ) = (x 2 + y 2 ) Example 1B: g(x, y ) = x 2 + y 2 Example 1C: h(x, y ) = x 2 y 2 Critical Thinking Questions In this section, you will examine the behavior of contour lines near points on which a function takes on local maximum and minimum values, and saddle points. (Q1) Look at the graph of f (x, y ) in Example 1A. At the origin, this function has a (circle one): local minimum local maximum saddle point (Q2) Look at the graph of g(x, y ) in Example 1B. At the origin, this function has a (circle one): local minimum local maximum saddle point (Q3) Look at the graph of h(x, y ) in Example 1C. At the origin, this function has a (circle one): local minimum local maximum saddle point (Q4) Describe the contour lines in Examples 1A and 1B, where there are local extrema (maximum and minimum values). How is the behavior of these contour lines near the origin different from the contour lines in Example 1C, where there is a saddle point?

4 Boise State Math 275 (Ultman) Worksheet 2.7: Local Extrema, Critical Points, 2 nd Derivative Test3 (Q5) On the contour map to the right, identify and label potential saddle points, and any points at which local extrema (maxima or minima) occur. Estimate the coordinates of these points. (You will find the exact points later in (Q14).) (Q6) On the contour map above, you should have identified one saddle point and one point at which an extremum occurs. Can you tell whether this extremum is a local maximum or a local minimum? If not, what additional information would you need in order to make this determination? ( Q7) The contour map used in questions (Q5,6) is generated by the function T (x, y) = x 3 +y 3 4xy. Given this additional information: (a) List at least three ways you can use this additional information to determine whether the extremum from (Q5) is a local maximum or local minimum. (b) Use one of the methods above to classify the extremum as a local minimum or local maximum. Which is it? ( Q8) Use a graphing app to graph the function T (x, y) = x 3 + y 3 4xy, and compare this graph to the contour map in (Q5). Suppose this function represents the temperature at each point (x, y) in the plane. Describe how the temperature changes for an ant as it moves away from the local minimum in different directions, and away from the saddle point in different directions.

5 Boise State Math 275 (Ultman) Worksheet 2.7: Local Extrema, Critical Points, 2 nd Derivative Test4 Model 2: Critical Points and Local Extrema & Saddle Points Example 2A: f (x, y) = 3 x 2 4 y 2 f (x, y) = x/2, 2y This function has a tangent plane when (x, y) = (0, 0). Example 2B: g(x, y) = x 2 + y 2 g(x, y) = x x 2 + y 2, y x 2 + y 2 This function does not have a tangent plane when (x, y) = (0, 0). Example 2C: h(x, y) = y 2 x 2 h(x, y) = 2x, 2y This function has a tangent plane when (x, y) = (0, 0). Critical Thinking Questions In this section, you will examine the connections between critical points, tangent planes, and local maxima, local minima, and saddle points. Definition: A function f (x, y) has a critical point at (a, b) if either: f (a, b) = 0 or f (a, b) does not exist f (a, b) = 0 when both partial derivatives are equal to zero: f x (a, b) = 0 and f y (a, b) = 0 f (a, b) does not exist when one (or both) partial derivatives do not exist: f x (a, b) does not exist and/or f y (a, b) does not exist

6 Boise State Math 275 (Ultman) Worksheet 2.7: Local Extrema, Critical Points, 2 nd Derivative Test5 (Q9) All three of the functions in Model 2 have a critical point at the same location. (a) What is the critical point for all three functions? (a, b) = (b) Circle the type of critical point each function has, and indicate the behavior of the tangent plane at the critical point. f (x, y ) (Example 2A) type of critical point: f (0, 0) = 0 / f (0, 0) d.n.e. tangent plane at critical point: is parallel to the xy-plane / does not exist g(x, y ) (Example 2B) type of critical point: g(0, 0) = 0 / g(0, 0) d.n.e. tangent plane at critical point: is parallel to the xy-plane / does not exist h(x, y ) (Example 2C) type of critical point: h(0, 0) = 0 / h(0, 0) d.n.e. tangent plane at critical point: is parallel to the xy-plane / does not exist (Q10) Each of the functions in Model 2 has either a local maximum, a local minimum, or a saddle point when (x, y) = (0, 0): f (x, y ) (Example 2A) has a (circle one): local minimum local maximum saddle point g(x, y ) (Example 2B) has a (circle one): local minimum local maximum saddle point h(x, y ) (Example 2C) has a (circle one): local minimum local maximum saddle point (Q11) Based on Model 2 and questions (Q9) and (Q10), indicate whether you agree or disagree with the following statement: Local maxima and local minima occur at critical points. If you disagree, modify the statement to make it true. (Q12) Based on Model 2 and questions (Q9) and (Q10), indicate whether you agree or disagree with the following statement: If (a, b) is a critical point of a function F (x, y), then F (a, b) will always be either a local maximum or a local minimum. If you disagree, modify the statement to make it true.

7 Boise State Math 275 (Ultman) Worksheet 2.7: Local Extrema, Critical Points, 2 nd Derivative Test6 (Q13) In Model 2, the functions in Examples A and C both have critical points that occur when f = 0. At these critical points, one has an extreme value (in this case, a local maximum), and the other has a saddle point. In these two examples, you can see that close to a local maximum (or minimum), the graph of the function is on one side of / passes through the tangent plane. On the other hand, close to a saddle point the graph of the function is on one side of / passes through the tangent plane. (This idea of comparing the behavior of the graph of a function near its tangent plane is used to define the curvature of a surface.) ( Q14) Go back to the function T (x, y) = x 3 + y 3 4xy from (Q5-8). Compute its gradient, and find the coordinates of the saddle point and the local minimum by solving the equation T (x, y) = 0.. ( Q15) Construct a function f (x, y) that has a critical point at the point (1, 2).

8 Boise State Math 275 (Ultman) Worksheet 2.7: Local Extrema, Critical Points, 2 nd Derivative Test7 Model 3: The Second Derivative Test The second derivative test is a tool for determining whether a critical point (a, b) of a function f (x, y) corresponds to a local maximum or a local minimum, or whether it is a saddle point. Conditions: To use the second derivative test, both of the following need to be true: i. f (a, b) = 0 ii. The first and second partial derivatives of f (x, y) must exist and be continuous near (a, b). The Test Function : The function used to test critical points is: D(x, y) = f xx (x, y)f yy (x, y) [ ] 2 f xy (x, y) or, in Leibniz notation: [ ] D(x, y) = 2 f 2 f 2 2 x 2 y f 2 y x How the Test Works: Step 1: Find all critical points (a, b) where f (a, b) = 0. Step 2: Compute the second partial derivatives of f, and set up the test function D(x, y). Step 3: Evaluate the test function D(x, y) at the critical points you found in Step 1. If D(a, b) > 0 and f xx (a, b) < 0, then f (a, b) is a local maximum. If D(a, b) > 0 and f xx (a, b) > 0, then f (a, b) is a local minimum. If D(a, b) < 0, then f has a saddle point at (a, b). Note: If D(a, b) = 0, the test is inconclusive. You need to find another way to determine whether f (a, b) is a local maximum, minimum, or saddle point.

9 Boise State Math 275 (Ultman) Worksheet 2.7: Local Extrema, Critical Points, 2 nd Derivative Test8 Critical Thinking Questions In this section, you will use the second derivative test to determine whether critical points correspond to local maxima or local minima, or whether they are saddle points. (Q16) If you completed (Q14), you should have found that the critical points of the temperature function T (x, y) = x 3 + y 3 4xy from (Q5-8) were (0, 0) and (4/3, 4/3). Use the second derivative test from Model 3 to show that (0, 0) is a saddle point, and (4/3, 4/3) is a local minimum of T. (Q17) For the function f (x, y) = x 2 + 2xy + y 3 : (a) Find the two critical points of f by solving the equation f (x, y) = 0. (b) Run the second derivative test on the points you found in part (a). Which point(s) (if any) correspond to local maxima, which (if any) correspond to local minima, and which (if any) are saddle points? Find the values for any local maxima or minima.

10 Boise State Math 275 (Ultman) Worksheet 2.7: Local Extrema, Critical Points, 2 nd Derivative Test9 ( Q18) In (Q9) and (Q10), you saw that the function g(x, y) = x 2 + y 2 has a critical point at (0, 0), corresponding to a local minimum (determined by looking at the graph of the function in Model 2, Example 2B). Using the second partial derivatives g xx, g yy, and g xy, eplain why you cannot use the second derivative test to show that this function has a local minimum at the origin. Summary For a function of two variables, f (x, y): Critical points occur when f = 0 (horizontal tangent plane), or when one or both of the partial derivatives does not exist (no tangent plane). Local maxima and minima occur only at critical points. Not every critical point corresponds to a local maximum or minimum. Saddle points can also occur at critical points. (Saddle points are two-dimensional versions of inflection points.) For a critical point (a, b) where f (a, b) = 0 and D(a, b) 0, the second derivative test will determine whether there is a local maximum, local minimum, or saddle point at P.