Math 1314 Lesson 24 Maxima and Minima of Functions of Several Variables

Transcription

1 Math 1314 Lesson 4 Maxima and Minima o Functions o Several Variables We learned to ind the maxima and minima o a unction o a single variable earlier in the course. We had a second derivative test to determine whether a critical point o a unction o a single variable generated a maximum or a minimum, or possibly that the test was not conclusive at that point. We will use a similar technique to ind relative extrema o a unction o several variables. Since the graphs o these unctions are more complicated, determining relative extrema is also more complicated. At a speciic critical number, we can have a max, a min, or something else. That something else is called a saddle point. The method or inding relative extrema is very similar to what you did earlier in the course. 1. Find the irst partial derivatives and set them equal to zero.. Find any points o intersection o the two equations in step 1 by using the intersect command. These points are your critical points. 3. Produce D( x, ( ). 4. Compute D or each critical point ( a,. Then you can apply the second derivative test or unctions o two variables: I D(a, > 0 and 0, then has a relative maximum at (a,. I D(a, > 0 and 0, then has a relative minimum at (a,. I D(a, < 0, then has a saddle point at (a,. I D(a, = 0, then this test is inconclusive. Lesson 4 Maxima and Minima o Functions o Several Variables 1

2 Recall: Produce D( x, ( ). Compute D or each critical point ( a,. Then you can apply the second derivative test or unctions o two variables: o I D(a, > 0 and 0, then has a relative maximum at (a,. o I D(a, > 0 and 0, then has a relative minimum at (a,. o I D(a, < 0, then has a saddle point at (a,. o I D(a, = 0, then this test is inconclusive. 3 3 Example 1: The critical points o (, ) x y 0.3 are (0, 0) and (0.1, 0.1). Classiy each critical point as a relative maximum, relative minimum or saddle point. For any relative maximum or relative minimum, calculate its maximum or minimum. D( x, ( ) Lesson 4 Maxima and Minima o Functions o Several Variables

3 Example : Find the relative extrema o the unction. Begin by entering the unction into GGB. a. Find the irst-order partials. x y x y (, ) 4 6 b. Find the point o intersection o the equations in part a. These points o intersection are the critical points o the unction. c. Determine D (, ). d. Apply the second derivative test to classiy each critical point ound in part b. RECALL: I D(a, > 0 and 0, then has a relative maximum at (a,. I D(a, > 0 and 0, then has a relative minimum at (a,. I D(a, < 0, then has a saddle point at (a,. I D(a, = 0, then this test is inconclusive. e. For any maxima point and minima point ound in part d, calculate the maxima and minima, respectively. Lesson 4 Maxima and Minima o Functions o Several Variables 3

4 Example 3: Suppose a company s weekly proits can be modeled by the unction P (, ) 0.x 0.5y x 90y 4000 where proits are given in thousand dollars and x and y denote the number o standard items and the number o deluxe items, respectively, that the company will produce and sell. How many o each type o item should be manuactured each week to maximize proit? What is the maximum proit that is realizable in this situation? Begin by entering the unction into GGB. a. Find the irst-order partials. b. Find the point o intersection o the equations in part a. These points o intersection are the critical points o the unction. c. Determine D (, ). d. Apply the second derivative test to classiy each critical point ound in part b. RECALL: I D(a, > 0 and 0, then has a relative maximum at (a,. I D(a, > 0 and 0, then has a relative minimum at (a,. I D(a, < 0, then has a saddle point at (a,. I D(a, = 0, then this test is inconclusive. Lesson 4 Maxima and Minima o Functions o Several Variables 4

5 e. How many o each type o item should be manuactured each week to maximize proit?. What is the maximum proit that is realizable in this situation? Example 4: The ollowing inormation or critical points. 3 3 (, ) 4 x y 3 is given below, classiy its Critical points: (0, 0) and (1, 1) x 3x 3y y 3y 3x 6x 6y 3 D(0,0) 9 D(1,1) 7 Lesson 4 Maxima and Minima o Functions o Several Variables 5