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sol2 - Partial Solutions for Sample Problems Two 1 Show...

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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 λ such that ∇ F = λ ∇ 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 θ = π/ 4; min F = 0 at r = 1 and θ = 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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sol2 - Partial Solutions for Sample Problems Two 1 Show...

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