1. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 62 § 7.3 Maxima and Minima of Functions of Several Variables

2. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 62  Relative Maxima and Minima  First Derivative Test for Functions of Two Variables  Second Derivative Test for Functions of Two Variables  Finding Relative Maxima and Minima Section Outline

3. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 62 Relative Maxima & Minima DefinitionExample Relative Maximum of f (x, y): f (x, y) has a relative maximum when x = a, y = b if f (x, y) is at most equal to f (a, b) whenever x is near a and y is is near b. Examples are forthcoming. DefinitionExample Relative Minimum of f (x, y): f (x, y) has a relative minimum when x = a, y = b if f (x, y) is at least equal to f (a, b) whenever x is near a and y is is near b. Examples are forthcoming.

4. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 62 First-Derivative Test If one or both of the partial derivatives does not exist, then there is no relative maximum or relative minimum.

5. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 62 Second-Derivative Test

6. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 62 Finding Relative Maxima & MinimaEXAMPLE SOLUTION Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. We first use the first-derivative test.

7. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 62 Finding Relative Maxima & Minima Now we set both partial derivatives equal to 0 and then solve each for y. CONTINUED Now we may set the equations equal to each other and solve for x.

8. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 62 Finding Relative Maxima & Minima We now determine the corresponding value of y by replacing x with 1 in the equation y = x + 2. CONTINUED So we now know that if there is a relative maximum or minimum for the function, it occurs at (1, 3). To determine more about this point, we employ the second-derivative test. To do so, we must first calculate

9. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 62 Finding Relative Maxima & Minima Since, we know, by the second-derivative test, that f (x, y) has a relative maximum at (1, 3). CONTINUED

10. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 62 Finding Relative Maxima & MinimaEXAMPLE SOLUTION A monopolist manufactures and sells two competing products, call them I and II, that cost $30 and $20 per unit, respectively, to produce. The revenue from marketing x units of product I and y units of product II is Find the values of x and y that maximize the monopolist’s profits. We first use the first-derivative test.

11. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 62 Finding Relative Maxima & Minima Now we set both partial derivatives equal to 0 and then solve each for y. CONTINUED Now we may set the equations equal to each other and solve for x.

12. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 12 of 62 Finding Relative Maxima & Minima We now determine the corresponding value of y by replacing x with 443 in the equation y = -0.1x + 280. CONTINUED So we now know that revenue is maximized at the point (443, 236). Let’s verify this using the second-derivative test. To do so, we must first calculate

13. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 13 of 62 Finding Relative Maxima & Minima Since, we know, by the second-derivative test, that R(x, y) has a relative maximum at (443, 236). CONTINUED