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Unformatted text preview: Partial Solutions for Sample Problems Two 1. Show that if P = ( x ,y ,z ) is a critical point of the function F ( P ) subject to the constraints G ( P ) = 0, and G z 6 = 0, then there exists such that F = G . 2. Find the critical points for the following functions and test whether each is a local maximum, minimum or saddle point: (a) x 4 + y 4 2 x 2 + 4 xy 2 y 2 (b) the function z(x,y) implicitly defined by x 2 + 2 y 2 + 3 z 2 2 xy 2 yz = 2. (a) x = 2, y = x , F = 4 is a minimum (b) x = 1 / 3, y = 2 x , z = 3 x are saddle points. 3. Find the extreme values of F ( x,y ) = 2 xy + (1 x 2 y 2 ) 1 2 in the region x 2 + y 2 1 using polar coordinates. max F = 5 / 4 at r = 3 / 2 and = / 4; min F = 0 at r = 1 and = 0. 4. Find the minimum value of x 3 + y 3 + z 3 for positive x,y,z subject to constraint ax + by + cz = 1, where a,b,c are positive. let d = a a + b b + c c . Then the max is 1 /d 2 when x = a/d , y = b/d and z = c/d . 5. Find the maximum and minimum values of x 2 + y 2 + z 2 subject to x 2 / 25 + y 2 / 25 + z 2 / 9 = 1 and x + y + 2 z = 0. min F = 225 / 11 at x = 33 / 15, y = x , z = x ; max F = 25 at x = 5 / 2, y = x , z = 0. Assume that f is a bounded function on rectangle R : 6. Define the notion of a partition P of [a,b] and define the corresponding lower and upper Riemann sums S P ( f ) and S P ( f ) as well the arbitrary Riemann sum S P ( P 1 ,...,P k ). Explain why S P ( f ) S P ( P 1 ,...,P,....
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This note was uploaded on 01/22/2012 for the course MAA 4103 taught by Professor Staff during the Fall '08 term at University of Florida.
 Fall '08
 Staff

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