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Analyzing Relative Maxima and Minima of Functions of Multiple Variables

This section delves into the concepts of relative maxima and minima for functions of two variables. It covers techniques such as the First Derivative Test, which helps identify points of interest where partial derivatives are zero. The Second Derivative Test is presented to classify these critical points further. Through examples, we solve for specific values that maximize profits in a monopolistic market, illustrating the practical applications of these mathematical concepts. The process for verification and addressing inconclusive tests is also highlighted.

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Analyzing Relative Maxima and Minima of Functions of Multiple Variables

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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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