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Unformatted text preview: Optimization John E. Gilbert, Heather Van Ligten, and Benni Goetz The surface to the right is the graph of z = f ( x, y ) = sin( x ) sin( y ) ,- π ≤ x, y ≤ π . It has mountains: Local Maxima as at Q , and it has basins: Local Minima as at P , both of which occured for curves. But it also has a pass through the mountains at which the terrain slopes up in one direction and down in another direction just like a saddle. So we also have to bring in the notion of Saddlepoint , corresponding to a pass through the mountains as at R . But, just as in the one variable case, everything centers on an algebraic and graphical understanding of local extrema and critical points. y z x P Q R S Definition: At ( a, b ) a function z = f ( x, y ) is said to have a • Local Maximum : f ( x, y ) ≤ f ( a, b ) for all ( x, y ) near ( a, b ) , • Local Minimum : f ( x, y ) ≥ f ( a, b ) for all ( x, y ) near ( a, b ) . • Saddle Point : if no matter how close we are to ( a, b ) , there are always points at which f ( x, y ) > f ( a, b ) and others at which f ( x, y ) < f ( a, b ) . The point ( a, b ) is said to be a Local Extremum of z = f ( x, y ) when one of these three is satisfied. In one variable locating local extrema usually meant finding where f ( x ) = 0 . In 2 variables we replace f ( x ) by ∇ f ( x, y ) . Definition: A point ( a, b ) is said to be a Critical Point of f ( x,y ) when ∇ f ( a, b ) = f x ( a, b ) i + f y ( a, b ) j = 0 , i.e. , f x ( a, b ) = f y ( a, b ) = 0 , or at least one of f x ( a, b ) , f y ( a, b ) does not exist. Two important observations: • if ( a, b ) is a local extremum, then ( a, b ) is a critical point, • if ∇ f ( a, b ) exists, then the tangent plane is horizontal at the point ( a, b, f ( a, b )) on the graph of z = f ( x, y ) when ( a, b ) is a critical point. The previous graph of z = sin( x ) sin( y ) shows that ∇ f ( a, b ) = 0 and the tangent plane is horizontal at P, Q, and R , whereas ∇ f ( a, b ) = 0 at S . Let’s see in detail how this works algebraically to find critical points: Start with the function z = f ( x, y ) = sin( x ) sin( y ) ,...
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- Spring '07
- Critical Point, Benni Goetz, John E. Gilbert