Extrema of Functions of Several Variables

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1 Extrema of Functions of Several Variables MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011

2 Background (1 of 3) In single-variable calculus there are three important results associated with locating the extrema of functions.

3 Background (1 of 3) In single-variable calculus there are three important results associated with locating the extrema of functions. Theorem Suppose that f is continuous on a closed, bounded interval [a, b]. The absolute extrema of f must occur at the endpoints of the interval (x = a or x = b) or at a critical number.

4 Background (2 of 3) Theorem (First Derivative Test) Suppose that f is continuous on interval [a, b] and c (a, b) is a critical number. 1 If f (x) > 0 for all x (a, c) and f (x) < 0 for all x (c, b) [in other words f changes from increasing to decreasing at c] then f (c) is a local maximum. 2 If f (x) < 0 for all x (a, c) and f (x) > 0 for all x (c, b) [in other words f changes from decreasing to increasing at c] then f (c) is a local minimum. 3 If f (x) has the same sign on (a, c) and (c, b), then f (c) is not a local extremum.

5 Background (3 of 3) Theorem (Second Derivative Test) Suppose f is continuous on the interval (a, b) and f (c) = 0 for some number c (a, b). 1 If f (c) < 0, then f (c) is a local maximum. 2 If f (c) > 0, then f (c) is a local minimum.

6 Local Extrema Definition We call f (a, b) a local maximum of f if there is an open disk R centered at (a, b), for which f (a, b) f (x, y) for all (x, y) in R. Similarly f (a, b) is a local minimum of f if there is an open disk R centered at (a, b), for which f (a, b) f (x, y) for all (x, y) in R. In either case, f (a, b) is called a local extremum of f.

7 Illustration f (x, y) = xye x 2 y y x

8 Critical Points Definition The point (a, b) is a critical point of the function f (x, y) if (a, b) is in the domain of f and either f f (a, b) = (a, b) = 0 x y or one or both of f / x and f / y do not exist at (a, b).

9 Critical Points Definition The point (a, b) is a critical point of the function f (x, y) if (a, b) is in the domain of f and either f f (a, b) = (a, b) = 0 x y or one or both of f / x and f / y do not exist at (a, b). Theorem If f (x, y) has a local extremum at (a, b), then (a, b) must be a critical point of f.

10 Example (1 of 2) Find the critical points of f (x, y) = x 2 + y 2 2x 6y y x

11 Example (2 of 2) 0 = f x = 2x 2 0 = f y = 2y 6 There is a unique critical point at (x, y) = (1, 3).

12 Saddle Points (1 of 4) Remark: to find extrema we will look for critical points; however, not all critical points are extrema. Consider the following function: f (x, y) = x 4 + y 4 4xy. Find the critical points, then examine the graph of the function.

13 Saddle Points (2 of 4) 0 = f x = 4x 3 4y 0 = f y = 4y 3 4x There are three critical points: (x 1, y 1 ) = ( 1, 1) (x 2, y 2 ) = (0, 0) (x 3, y 3 ) = (1, 1)

14 Saddle Points (3 of 4) 2 1 y z y x 0 x

15 Saddle Points (4 of 4) Definition The point P(a, b, f (a, b)) is a saddle point of z = f (x, y) if (a, b) is a critical point of f and if every open disk centered at (a, b) contains points (x, y) in the domain of f for which f (x, y) < f (a, b) and points (x, y) in the domain of f for which f (x, y) > f (a, b).

16 Second Derivatives Test Theorem (Second Derivatives Test) Suppose that f (x, y) has continuous second-order partial derivatives in some open disk containing the point (a, b) and that f x (a, b) = f y (a, b) = 0. Define the discriminant D(a, b) by D(a, b) = f xx (a, b)f yy (a, b) [f xy (a, b)] 2. 1 If D(a, b) > 0 and f xx (a, b) > 0, then f has a local minimum at (a, b). 2 If D(a, b) > 0 and f xx (a, b) < 0, then f has a local maximum at (a, b). 3 If D(a, b) < 0, then f has a saddle point at (a, b). 4 If D(a, b) = 0, then no conclusion can be drawn.

17 Examples (1 of 2) Find the extrema and saddle points of f (x, y) = x 4 + y 4 4xy.

18 Examples (1 of 2) Find the extrema and saddle points of f (x, y) = x 4 + y 4 4xy. D(x, y) = (12x 2 )(12y 2 ) ( 4) 2 = 144x 2 y 2 16 Using the critical points found earlier, D( 1, 1) = 128 > 0 and f xx ( 1, 1) = 12 > 0 = local minim D(0, 0) = 16 < 0 = saddle point D(1, 1) = 128 > 0 and f xx (1, 1) = 12 > 0 = local minimum

19 Examples (2 of 2) Find the extrema and saddle points (if any) of f (x, y) = 2x 2 + y 3 x 2 y 3y.

20 Examples (2 of 2) Find the extrema and saddle points (if any) of f (x, y) = 2x 2 + y 3 x 2 y 3y. 0 = f x = 4x 2xy 0 = f y = 3y 2 x 2 3 Critical points: (0, 1), (0, 1), ( 3, 2), (3, 2) Discriminant: D(x, y) = 4x 2 12y y D(0, 1) = 12 > 0 and f xx (0, 1) = 2 > 0 = local minimum D(0, 1) = 36 < 0 = saddle point D( 3, 2) = 36 < 0 = saddle point D(3, 2) = 36 < 0 = saddle point

21 Example (1 of 2) Find the points on the surface z 2 = x y + 1 that are closest to the origin.

22 Example (1 of 2) Find the points on the surface z 2 = x y + 1 that are closest to the origin. Let f (x, y) be the squared distance from (x, y, z) to the origin. f (x, y) = (x 0) 2 +(y 0) 2 +(z 0) 2 = x 2 +y 2 +z 2 = x 2 +y 2 +x y+1

23 Example (2 of 2) 2 y z x 1 J. Robert Buchanan 2 Extrema of Functions of Several Variables

24 Linear Regression/Least Squares A common method of data analysis is linear regression, the process of finding the best fitting line through a collection of ordered pairs x

25 Error Suppose the data consists of ordered pairs {(x 1, y 1 ), (x 2, y 2 ),..., (x n, y n )}.

26 Error Suppose the data consists of ordered pairs {(x 1, y 1 ), (x 2, y 2 ),..., (x n, y n )}. Assume the linear fit is given by y = ax + b.

27 Error Suppose the data consists of ordered pairs {(x 1, y 1 ), (x 2, y 2 ),..., (x n, y n )}. Assume the linear fit is given by y = ax + b. Define e i = y i (ax i + b).

28 Error Suppose the data consists of ordered pairs {(x 1, y 1 ), (x 2, y 2 ),..., (x n, y n )}. Assume the linear fit is given by y = ax + b. Define e i = y i (ax i + b). Determine the values of a and b which minimize f (a, b) = n ei 2 = i=1 n [y i (ax i + b)] 2. i=1

29 Critical Points f a (a, b) = 2 f b (a, b) = 2 n (y i (ax i + b))x i i=1 n (y i (ax i + b)) i=1

30 Critical Points f a (a, b) = 2 f b (a, b) = 2 n (y i (ax i + b))x i i=1 n (y i (ax i + b)) i=1 Thus the critical point is the solution to following system of two equations. a n i=1 x 2 i a + b n x i = i=1 n x i + bn = i=1 n x i y i i=1 n i=1 y i

31 Result

32 Example (1 of 2) Consider the data set {( 1, 6), (1, 5), (2, 3), (5, 3), (7, 1)} and find the best fit line for these points x

33 Example (2 of 2) x i y i Then the system of equations n n a + b x i = can be written as i=1 x 2 i a i=1 n x i + bn = i=1 n x i y i i=1 n i=1 a(80) + b(14) = 13 a(14) + b(5) = 16. Thus (a, b) = ( 53/68, 183/34) and the best fitting line has equation y = x y i

34 Method of Steepest Ascent Recall: the gradient points in the direction of greatest increase for a function. Given z = f (x, y), let (x 0, y 0 ) be an initial approximation to the location of a local maximum, calculate f (x 0, y 0 ) which will point in the direction of greatest increase for f, move along the parametric curve until f stops increasing r(t) = x 0, y 0 + f (x 0, y 0 )t call this point (x 1, y 1 ) and repeat until we reach a maximum.

35 Example (1 of 4) 3y Let f (x, y) = x 2 + y 2 + 1, then f (x, y) = 6xy (x 2 + y 2 + 1) 2, 3(1 + x 2 y 2 ) (x 2 + y 2 + 1) 2. x z y 5

36 Example (2 of 4) Choose (x 0, y 0 ) = ( 2, 1), then f ( 2, 1) = 1/3, 1/3. Parametric line: x(t) = 2 t/3 y(t) = 1 t/3 Find the maximum of f (x(t), y(t)) = f ( 2 t/3, 1 t/3) = 9(t 3) 2(t 2 + 3t + 27). When t , (x 1, y 1 ) ( , ).

37 Example (3 of 4) x z y 5

38 Example (4 of 4) 1 z y 0 x 5 5

39 Absolute Extrema Definition We call f (a, b) the absolute maximum of f on the region R if f (a, b) f (x, y) for all (x, y) in R. Similarly, f (a, b) the absolute minimum of f on the region R if f (a, b) f (x, y) for all (x, y) in R. In either case, f (a, b) is called an absolute extremum of f.

40 Absolute Extrema Definition We call f (a, b) the absolute maximum of f on the region R if f (a, b) f (x, y) for all (x, y) in R. Similarly, f (a, b) the absolute minimum of f on the region R if f (a, b) f (x, y) for all (x, y) in R. In either case, f (a, b) is called an absolute extremum of f. Definition A subset R of the xy-plane is bounded if there is a disk that completely contains R.

41 Extreme Value Theorem Theorem (Extreme Value Theorem) Suppose that f (x, y) is continuous on the closed and bounded subset R of the xy-plane. Then f has both an absolute maximum and an absolute minimum on R. Further, an absolute extremum may only occur at a critical point in R or at a point on the boundary of R.

42 Example Find the absolute extrema of f (x, y) = x 2 + y 2 2x 4y on R, a triangle in the xy-plane with vertices at (0, 0), (0, 3), and (3, 3) x z y 1 0

43 Homework Read Section Exercises: 1 7 odd, 23, 29, 31, odd

critical points: 1,1, 1, 1, 1,1, 1, 1

critical points: 1,1, 1, 1, 1,1, 1, 1 Extrema of Functions of Several Variables -(.7). Local Extremes and Critical Points Definition: The point a,b is a critical point of a function f x,y if a,b is in the domain of f and either f a,b or f

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