Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds

CMIIh 2020-12-24 Answered
Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points.
f(x,y)=yexey
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Expert Answer

joshyoung05M
Answered 2020-12-25 Author has 97 answers

To find the critical point, find fx,fy and set fx=0,fy=0
fx=tex=0y=0 or ex=0
For ex=0 have no solution for xR
Therefore y=0
Set fy=0
fy=exey=0
Substitute y=0,fy=exe0=0ex1=0ex=1
Taking ln on both sides, ln(ex)=ln(1) , x=0
Thus (0,0) is the ritical point.
D(x,y)=f×(x,y)fyy(x,y)(fxy(x,y))2
1) If D(a,b)>0 and f×(a,b)>0 then (a,b) is local aximum of f
2) If D(a,b)>0 and f×(a,b)<0 then (a,b) is local aximum of f
3) If D(a,b)<0 then f(a,b) is saddle point
4) If D(a,b)=0 then this is incobclusive
Here, f×=yex,fyy=ey and fxy=ex
D(x,y)=tex(ey)(ex)2=eyx+ye2x
The critical point is(0,0)
D(0,0)=0e0e20=e0=1
Thus D(0,0)=-1<0 then f(0,0) is saddle point,
f(0,0)=(0)e0e0=e0=1
Therefore -1 is saddle point.

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