Question
Asked Nov 12, 2019
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Let f(x,y)= x2+y2+kxy

a) Show that if k<-2 or k>, the origin (0,0) is a saddle point.

b) Suppose -2<k<2. Is the origin a local max., min., or a saddle point? And, when k= 2, is the origin local max., min., or a saddle point?

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f(x, y) x2 +y y Consider the given function as Theorem used: Supposed that the function f(x,y) and its first and second derivative is continuous throughout at disk cantered at (a,b) and f(a,b) f(a,b) = 0, then following conditions are true. 1.f has a local maximum at (a,b) if f <0 andf- f0 2.f has a local minimum at (ab) if f >0 andf-f>0 xxyy 3.f has a saddle point at (a,b) if ff-0 4.f has a no conclusion at (a,b) if f- 0 Calculate the partial derivative of the function f(x, y) =x + ykxy with respect to x as follows. (x2ykxy2x +ky ax Calculate the partial derivative of the function f(x,y) with respect to y as follows. -(x2 + y +kxy)=2y +kx

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Calculate the second order partial derivative of the function f(x,y) with respect to x as follows. a(of axax (2x+ ky) 2 Calculate the second order partial derivative of the function f(x,y)with respect to y as follows aaf (2y+kx) 2 Calculate the mixed partial derivative of the function f(x,y) with respect to x and y as follows. aof (2x+ky)-k Calculate the mixed partial derivative of the function f (x.y)with respect to y and x as follows. aaf (2y+kx) k Calculate the critical point by equating f -0 and f-0 and solve for x and y. 0 2x+ky 0 kr 2y 0 by solving these equation, we get (x. y) = (0,0)

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