Maxima and Minima. Marius Ionescu. November 5, Marius Ionescu () Maxima and Minima November 5, / 7


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1 Maxima and Minima Marius Ionescu November 5, 2012 Marius Ionescu () Maxima and Minima November 5, / 7
2 Second Derivative Test Fact Suppose the second partial derivatives of f are continuous on a disk with center (a, b), and suppose that f x (a, b) = 0 and f y (a, b) = 0. Let D = f xx f yx f xy f yy = f xx f yy (f xy ) 2. 1 If D > 0 and f xx (a, b) > 0, then f (a, b) is a local minimum. 2 If D > 0 and f xx (a, b) < 0, then f (a, b) is a local maximum. 3 If D < 0, then f (a, b) is not a local maximum or minimum. In this case the point (a, b) is called a saddle point of f. Marius Ionescu () Maxima and Minima November 5, / 7
3 Examples Examples Marius Ionescu () Maxima and Minima November 5, / 7
4 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Marius Ionescu () Maxima and Minima November 5, / 7
5 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Find the points on the surface x 2 y 2 z = 1 that are closest to the origin. Marius Ionescu () Maxima and Minima November 5, / 7
6 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Find the points on the surface x 2 y 2 z = 1 that are closest to the origin. Find three positive numbers x, y, and z whose sum is 100 and whose product is maximum. Marius Ionescu () Maxima and Minima November 5, / 7
7 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Find the points on the surface x 2 y 2 z = 1 that are closest to the origin. Find three positive numbers x, y, and z whose sum is 100 and whose product is maximum. A rectangular box without a lid is to be made from 12 m 2 of cardboard. Find the maximum volume of such a box. Marius Ionescu () Maxima and Minima November 5, / 7
8 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Find the points on the surface x 2 y 2 z = 1 that are closest to the origin. Find three positive numbers x, y, and z whose sum is 100 and whose product is maximum. A rectangular box without a lid is to be made from 12 m 2 of cardboard. Find the maximum volume of such a box. Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid 9x y 2 + 4z 2 = 36. Marius Ionescu () Maxima and Minima November 5, / 7
9 Absolute Maximum and Minimum Values Denition Marius Ionescu () Maxima and Minima November 5, / 7
10 Absolute Maximum and Minimum Values Denition A closed set in R 2 is one that contains all its boundary points. Marius Ionescu () Maxima and Minima November 5, / 7
11 Absolute Maximum and Minimum Values Denition A closed set in R 2 is one that contains all its boundary points. A bounded set in R 2 is one that is contained within some disk. Marius Ionescu () Maxima and Minima November 5, / 7
12 Extreme Value Theorem for Functions of Two Variables Theorem If f is continuous on a closed, bounded set D in R 2, then f attains an absolute maximum value f (x 1, y 1 ) and an absolute minimum value f (x 2, y 2 ) at some points (x 1, y 1 ) and (x 2, y 2 ) in D. Marius Ionescu () Maxima and Minima November 5, / 7
13 Extension of the Closed Interval Method Fact To nd the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: Marius Ionescu () Maxima and Minima November 5, / 7
14 Extension of the Closed Interval Method Fact To nd the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: 1 Find the values of f at the critical points of f in D. Marius Ionescu () Maxima and Minima November 5, / 7
15 Extension of the Closed Interval Method Fact To nd the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: 1 Find the values of f at the critical points of f in D. 2 Find the extreme values of f on the boundary of D. Marius Ionescu () Maxima and Minima November 5, / 7
16 Extension of the Closed Interval Method Fact To nd the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: 1 Find the values of f at the critical points of f in D. 2 Find the extreme values of f on the boundary of D. 3 The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value. Marius Ionescu () Maxima and Minima November 5, / 7
17 Examples Examples Marius Ionescu () Maxima and Minima November 5, / 7
18 Examples Examples Find the absolute maximum and minimum values of the function f (x.y) = x 2 2xy + 2y on the rectangle D = {(x, y) 0 x 3, 0 y 2}. Marius Ionescu () Maxima and Minima November 5, / 7
19 Examples Examples Find the absolute maximum and minimum values of the function f (x.y) = x 2 2xy + 2y on the rectangle D = {(x, y) 0 x 3, 0 y 2}. Find the absolute maximum and minimum values of the function f (x, y) = 3 + xy x 2y on the closed triangular region with vertices (1, 0), (5, 0), and (1, 4). Marius Ionescu () Maxima and Minima November 5, / 7
14.7: Maxima and Minima
14.7: Maxima and Minima Marius Ionescu October 29, 2012 Marius Ionescu () 14.7: Maxima and Minima October 29, 2012 1 / 13 Local Maximum and Local Minimum Denition Marius Ionescu () 14.7: Maxima and Minima
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