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Unformatted text preview: Unit 8 Optimization In your introductory calculus course, you will have learned how the first and second derivatives of a function y = f ( x ) give information about maxima and minima (i.e. extreme values) of the function. We know that a univariate function y = f ( x ) (i.e. a function of only 1 independent variable) can only have a local maximum or a local minimum at a critical number in the domain of f , where c is a critical number of f if either f prime ( c ) = 0 or f prime ( c ) does not exist. Furthermore, for a continuous function f , if f prime ( c ) = 0 and f primeprime ( c ) > then f has a local minimum at x = c , while if f prime ( c ) = 0 and f primeprime ( c ) < 0 then f has a local maximum at c . ( Recall: f prime ( c ) = 0 means that f ( x ) has a horizontal tangent line at x = c . f primeprime ( c ) > 0 means that f is concave up at x = c , so the horizontal tangent indicates that f was decreasing to the left of x = c and starts increasing to the right of x = c , having reached a local minimum at x = c . Similarly, f primeprime ( c ) < 0 means that f is concave down at x = c , so in this case the hori zontal tangent indicates that f was increasing to the left of x = c and starts decreasing to the right of x = c , having reached a local maximum at x = c .) We have similar properties with functions of 2 variables. Of course, since a surface z = f ( x, y ) is more complicated than a curve y = f ( x ), some of the definitions and results are a bit more complex. Definition 8.1. Let D be the domain of a function f ( x, y ) and let ( a, b ) ∈ D . Then we say that (i) f ( a, b ) is the maximum value of f ( x, y ) on D if f ( a, b ) ≥ f ( x, y ) for every ( x, y ) ∈ D . (ii) f ( a, b ) is the minimum value of f ( x, y ) on D if f ( a, b ) ≤ f ( x, y ) for every ( x, y ) ∈ D . 1 Note: The maximum and minimum values of f ( x, y ) are sometimes referred to as the absolute maximimum and absolute minimum values, respectively. Definition 8.2. The extreme values of f ( x, y ) are the absolute minimum and absolute maximum values of the function. If f ( a, b ) is not the largest (or smallest) value of f that occurs anywhere in the domain, but is the largest (or smallest) value that occurs anywhere near this point (i.e. in the neighbourhood of ( a, b )), then f has a local max imum (or minimum) at ( a, b ). Definition 8.3. Let D be the domain of a function f ( x, y ) and let ( a, b ) ∈ D . Then we say that (i) f ( a, b ) is a local maximum value of f ( x, y ) if f ( a, b ) ≥ f ( x, y ) for every ( x, y ) nearby ( a, b ). (ii) f ( a, b ) is a local minimum value of f ( x, y ) if f ( a, b ) ≤ f ( x, y ) for every ( x, y ) nearby ( a, b ). Figure 1 shows part of the surface f ( x, y ) = x 2 + y 2 which is easily seen to have both a local minimum and a minimum (ie. absolute minimum) at the point (0 , 0), so the minimum value of f ( x, y ) = x 2 + y 2 is f (0 , 0) = 0....
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This note was uploaded on 11/03/2009 for the course MATH 0110A taught by Professor Olds,vicky during the Fall '09 term at UWO.
 Fall '09
 OLDS,VICKY
 Calculus, Derivative

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