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Maxima and Minima of Functions: Second-Derivative Test Explained

Learn how to find relative maxima and minima using the second-derivative test for functions of two variables. Detailed examples and solutions included.

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Maxima and Minima of Functions: Second-Derivative Test Explained

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  1. §7.3 Maxima and Minima of Functions of Several Variables

  2. Section Outline • 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

  3. Relative Maxima & Minima

  4. First-Derivative Test If one or both of the partial derivatives does not exist, then there is no relative maximum or relative minimum.

  5. Second-Derivative Test

  6. Finding Relative Maxima & Minima EXAMPLE 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. SOLUTION We first use the first-derivative test.

  7. Finding Relative Maxima & Minima CONTINUED Now we set both partial derivatives equal to 0 and then solve each for y. Now we may set the equations equal to each other and solve for x.

  8. Finding Relative Maxima & Minima CONTINUED We now determine the corresponding value of y by replacing x with 1 in the equation y = x + 2. 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. Finding Relative Maxima & Minima CONTINUED Since , we know, by the second-derivative test, that f(x, y) has a relative maximum at (1, 3).

  10. Finding Relative Maxima & Minima EXAMPLE 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. SOLUTION We first use the first-derivative test.

  11. Finding Relative Maxima & Minima CONTINUED Now we set both partial derivatives equal to 0 and then solve each for y. Now we may set the equations equal to each other and solve for x.

  12. Finding Relative Maxima & Minima CONTINUED We now determine the corresponding value of y by replacing x with 443 in the equation y = -0.1x + 280. 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. Finding Relative Maxima & Minima CONTINUED Since , we know, by the second-derivative test, that R(x, y) has a relative maximum at (443, 236).

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