Transcription of SOLUTIONS - UCSD Mathematics
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SOLUTIONS - UCSD Mathematics
SOLUTIONSP roblem the critical points of the functionf(x,y) = 2x3 3x2y 12x2 3y2and determine their type local min/local max/saddle point. Are there any global min/max?Solution:Partial derivativesfx= 6x2 6xy 24x,fy= 3x2 find the critical points, we solvefx= 0 = x2 xy 4x= 0 = x(x y 4) = 0 = x= 0orx y 4 = 0fy= 0 = x2+ 2y= 0we findy= 0from the second equation. In the second case, we solve the systembelow by substitutionx y 4 = 0,x2+ 2y= 0 = x2+ 2x 8 = 0= x= 2orx= 4 = y= 2ory= three critical points are(0,0),(2, 2),( 4, 8).To find the nature of the critical points, we apply the second derivative test. We haveA=fxx= 12x 6y 24, B=fxy= 6x, C=fyy= the point(0,0)we havefxx= 24,fxy= 0,fyy= 6 = AC B2= ( 24)( 6) 0>0 = (0,0)is local , we find(2, 2)is a saddle pointsinceAC B2= (12)( 6) ( 12)2=<0and( 4, 8)is saddlesinceAC B2= ( 24)( 6) (24)2< function has no global min sincelimy ,x=0f(x,y) = and similarly there is no global maximum sincelimx ,y=0f(x,y) =.
SOLUTIONS Problem 1. Find the critical points of the function f(x;y) = 2x3 3x2y 12x2 3y2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max?
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