1. Findthepartialderivativesof

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f(x,y)=

x2 −1

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xy

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with respect to x and y, where f is defined on the positive quadrant

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 R
 2

++ = {(x, y) : x >0, y >0}.

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1. Let

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wherex>0.

f(x, y)= x(y2 −xy + 1)

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1. Show that the set of points (x, y) suchthat

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f(x, y)= 2

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DOES NOT defines a function y = g(x) locally around x = 2.

1. Considerthesetofpoints(x,y)suchthat

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f(x, y)= 2

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and that y >1. Show that the set defines a function y = g(x) locally around x = 2 by plotting the set. (You can use https://www.desmos.com/calculator or other softwares.)

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1. Find gl(x) evaluated at x = 2 in part(b).

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1. Letf(x,y)=min{x,y},definedonthex≥0,y≥0,wheremin{x,y}isthefunction thatspitsthesmallerofthetwonumbersxand(e.g.min{1,3}=1,min{3,2}= 2.) Show that fis not differentiable at any point on the 45-degree line. (Hint: see thelecturenoteormypreviouspostonPiazzaforexamples)

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1. Solve the maximizationproblem

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max

(x,y)∈A

x2 −y2

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where A = [−1, 1] × [−1, 1]. Is there a point in A such that fx = fy = 0? Is that point a maximum?

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1. Give an algorithm to solve the minimization problem

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min

(x,y)∈[0,1]×[0,1]

f(x,y)

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wherefisdifferentiableon[0,1]×[0,1].(Aminimizationproblemasksyoutofind the point (x, y) in A that gives the lowest function value than other points inA.)

P(5.u)

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