F(x,y)=x3+y3−63(x+y)+12xy
0 It's a polynomial function with continuous partial derivatives of any order.
1 What are the critical points?
Fx′=3x2−63+12y
Fy′=3y2−63+12x
x2−21+4y=0
y2−21+4x=0
2 Can we solve the system?
4y=21−x2(21−x2)2−21∗16+4∗16∗x=0
441−42x2+x4−336+64x=0
105−42x2+x4+64x=0
x4−42x2+64x+105=0:=P(x)
105=1∗3∗5∗7
P(1)=0, P(−1)=0, P(3)=0, P(−3)=0
P(5)=0, P(−5)=0, P(7)=0, P(−7)=0
Good, all roots (pairs of) are real (and symmetric).
x=−7 x=−1 x=3 x=5
y=−7 y=5 y=3 y=−1
3 What does the second derivative test say?
D=Fxx′′∗Fxx′′−Fxy′′∗Fxy′′
Fxx′′=6x
Fyy′′=6y
Fxy′′=12
D=36xy−144
D(−7,−7)>0, Fxx′′<0⟹local maximum
D(−1,5)<0 ⟹saddle point
D(3,3)>0, Fxx′′>0⟹local minimum
D(5,−1)<0 ⟹saddle point
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