Find the critical points of the following functions. f(x,\ y)=ye^{x}-e^{y}

Trent Carpenter

Trent Carpenter

Answered question

2021-10-09

Find the critical points of the following functions.
f(x, y)=yexey

Answer & Explanation

Isma Jimenez

Isma Jimenez

Skilled2021-10-10Added 84 answers

Step 1
To find the critical point, find fx, fy and set fx=0, fy=0
fx=yex
fy=exey
Now set fx=0
fx=yex=0
y=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=0
ex1=0
ex=1
Taking ln on both sides,
ln(ex)=ln(1)
x=0
Thus (0, 0) is the crirtical point.
Step 2
D(x, y)=f×(x, y)fyy(x, y)(fxy(x, y))2
1) If D(a, b)>0 amd f×(a, b)>0 then (a, b) is local minimum of f
2) If D(a, b)>0 and f×(a, b)<0 then (a, b) is local maximum of f
3) If D(a, b)<0 then f(a, b) is saddle point.
4) If D(a, b)=0 then this test is inconclusive.
Here f×=yex, fyy=ey and fxy=ex
D(x, y)=yex(ey)(ex)2
=yex+ye2x
The critical point is (0, 0)

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