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.
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. …
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