Saddle point: An example is the point 0 , 0 for the...

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Lecture 10 Optimization problems for multivariable functions Local maxima and minima - Critical points (Relevant section from the textbook by Stewart: 14.7) Our goal is to now find maximum and/or minimum values of functions of several variables, e.g., f ( x, y ) over prescribed domains. As in the case of single-variable functions, we must first establish the notion of critical points of such functions. Recall that a critical point of a function f ( x ) of a single real variable is a point x for which either (i) f ( x ) = 0 or (ii) f ( x ) is undefined. Critical points are possible candidates for points at which f ( x ) attains a maximum or minimum value over an interval. Also recall that if f ( x ) = 0, it could be a (i) local minimum, (ii) local maximum or (iii) point of inflection. We can determine the nature of this critical point from a look at f ′′ ( x ), provided it exists. Up to now, we have encountered three types of critical points for functions f ( x, y ) of two variables: 1. Local minima: The point (0 , 0) is a local minimum for the function f ( x, y ) = x 2 + y 2 , the graph of which is sketched below. O x y z z = x 2 + y 2 A plot of the countours/level sets of this function will also help us to understand the behaviour of this function around its local minimum. Such a plot, originally presented in Lecture 4, is 67
Some level sets of z = x 2 + y 2 x y 0 1 2 -1 -2 C = 1 C = 4 C = 0 shown again below. The level sets of f ( x, y ) satisfy the equation, x 2 + y 2 = C. (1) As such, they are concentric circles of radius C centered at (0 , 0). As C approaches zero from above, these circles get smaller. The level set corresponding to C = 0 is the point (0 , 0), which represents the minimum value of f achieved at (0 , 0). The definition of a local minimum seems quite straightforward but we state it here for the sake of completeness. (You’ll also find it in the textbook.) A point ( a, b ) is a local minimum of the function f ( x, y ) if there exists a circle C r of radius r > 0 centered at ( a, b ) such that f ( x, y ) f ( a, b ) for all ( x, y ) lying inside C r . (2) Notes: In many books, the term “relative minimum” is used instead of “local minimum.” The exact radius r of the circle is not important here. What is important is that a circular region of radius r > 0 exists. 2. Local maxima: The point (0 , 0) is a local maximum for the function f ( x, y ) = 50 x 2 2 y 2 , the graph of which is sketched below. (This was the hotplate function studied earlier.) Once again, a plot of the contours for this function may be helpful to see how they get smaller and converge toward the single point at (0 , 0) which now represents a local maximum: 68
50 x y z O z = 50 - x 2 - 2 y 2 x y C = 50 41 46 1 2 3 4 49 34 A point ( a, b ) is a local maximum of the function f ( x, y ) if there exists a circle C r of radius r > 0 centered at ( a, b ) such that f ( x, y ) f ( a, b ) for all ( x, y ) lying inside C r . (3) Note: In many books, the term “relative maximum” is used instead of “local maximum.” There is one other special kind of critical point: 3. Saddle point: An example is the point (0 , 0) for the function f ( x, y ) = x 2 y 2 . We sketch a graph of f near (0 , 0). Two noteworthy points can be made from this graph: (a) Consider the set of points ( x, y ) = ( x,

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