Differentiating g leads to g ′ ( t ) = 2 t − 4. Define g ( t ) = f ( x ( t ), y ( t ) ). L 1 L 1 is the line segment connecting ( 0, 0 ) ( 0, 0 ) and ( 4, 0 ), ( 4, 0 ), and it can be parameterized by the equations x ( t ) = t, y ( t ) = 0 x ( t ) = t, y ( t ) = 0 for 0 ≤ t ≤ 4. The method of Lagrange multipliers is introduced in Lagrange Multipliers.įigure 4.52 Graph of the domain of the function f ( x, y ) = x 2 − 2 x y + 4 y 2 − 4 x − 2 y + 24. If the boundary of the set D D is a more complicated curve defined by a function g ( x, y ) = c g ( x, y ) = c for some constant c, c, and the first-order partial derivatives of g g exist, then the method of Lagrange multipliers can prove useful for determining the extrema of f f on the boundary. The same approach can be used for other shapes such as circles and ellipses. If the boundary is a rectangle or set of straight lines, then it is possible to parameterize the line segments and determine the maxima on each of these segments, as seen in Example 4.40. įinding the maximum and minimum values of f f on the boundary of D D can be challenging. The maximum and minimum values of f f will occur at one of the values obtained in steps 2 and 3.Determine the maximum and minimum values of f f on the boundary of its domain.Calculate f f at each of these critical points.Determine the critical points of f f in D.To find the absolute maximum and minimum values of f f on D, D, do the following: Let z = f ( x, y ) z = f ( x, y ) be a continuous function of two variables defined on a closed, bounded set D, D, and assume f f is differentiable on D. Problem-Solving Strategy: Finding Absolute Maximum and Minimum Values Therefore the only possible values for the global extrema of f f on D D are the extreme values of f f on the interior or boundary of D. But an interior point ( x 0, y 0 ) ( x 0, y 0 ) of D D that’s an absolute extremum is also a local extremum hence, ( x 0, y 0 ) ( x 0, y 0 ) is a critical point of f f by Fermat’s theorem. In particular, if either extremum is not located on the boundary of D, D, then it is located at an interior point of D. The proof of this theorem is a direct consequence of the extreme value theorem and Fermat’s theorem. The values of f f on the boundary of D.The values of f f at the critical points of f f in D.Then f f will attain the absolute maximum value and the absolute minimum value, which are, respectively, the largest and smallest values found among the following: Apply Second Derivative Test to determine whether each critical point is a local maximum, local minimum, or saddle point, or whether the theorem is inconclusive.įinding Extreme Values of a Function of Two VariablesĪssume z = f ( x, y ) z = f ( x, y ) is a differentiable function of two variables defined on a closed, bounded set D.Calculate the discriminant D = f x x ( x 0, y 0 ) f y y ( x 0, y 0 ) − ( f x y ( x 0, y 0 ) ) 2 D = f x x ( x 0, y 0 ) f y y ( x 0, y 0 ) − ( f x y ( x 0, y 0 ) ) 2 for each critical point of f.Discard any points where at least one of the partial derivatives does not exist. Determine the critical points ( x 0, y 0 ) ( x 0, y 0 ) of the function f f where f x ( x 0, y 0 ) = f y ( x 0, y 0 ) = 0.To apply the second derivative test to find local extrema, use the following steps: Let z = f ( x, y ) z = f ( x, y ) be a function of two variables for which the first- and second-order partial derivatives are continuous on some disk containing the point ( x 0, y 0 ). Problem-Solving Strategy: Using the Second Derivative Test for Functions of Two Variables For functions of two or more variables, the concept is essentially the same, except for the fact that we are now working with partial derivatives. Critical Pointsįor functions of a single variable, we defined critical points as the values of the function when the derivative equals zero or does not exist. The main ideas of finding critical points and using derivative tests are still valid, but new wrinkles appear when assessing the results. This application is also important for functions of two or more variables, but as we have seen in earlier sections of this chapter, the introduction of more independent variables leads to more possible outcomes for the calculations. One of the most useful applications for derivatives of a function of one variable is the determination of maximum and/or minimum values. 4.7.3 Examine critical points and boundary points to find absolute maximum and minimum values for a function of two variables.4.7.2 Apply a second derivative test to identify a critical point as a local maximum, local minimum, or saddle point for a function of two variables.4.7.1 Use partial derivatives to locate critical points for a function of two variables.
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