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Unformatted text preview: Midterm Solutions Derivatives 1. For each function, decide whether the graph has at least one local max, at least one local min, or neither. (A point ( p,q ) is a local max for f if f ( x, y ) f ( p,q ) for all ( x,y ) sufficiently near ( p,q ) . ) (a) f ( x,y ) = x 2- y 2 Circle one: Local max Local Min Neither Solution: Simply look at the graph which is a saddle to see there are neither local minima nor local maxima. You may also use the second derivative test. (b) f ( x,y ) = 3 x 2 +2 y 2 Local max Local Min Neither Solution: The graph is an elliptical cup and so has a local minimum. You may also apply the second derivative test. (c) f ( x,y ) =- x- y Local max Local Min Neither Solution: Neither. This is a plane that has no (local) maximum or minimum points. (d) f ( x,y ) = ( x- 1) 2 Local max Local Min Neither Solution: The graph is a trough. Every point along the bottom of the trough of the form (1,y) is a local minimum....
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- Spring '06