Find the critical points of the following functions. f(x, y)=yex−ey

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Find the critical points of the following functions.  f(x, y)=yex−ey

Isma Jimenez · Accepted Answer

Step 1To find the critical point, find fx, fy and set fx=0, fy=0fx=yexfy=ex−eyNow set fx=0fx=yex=0⇒y=0 or ex=0For ex=0 have no solution for x∈RTherefore y=0Set fy=0fy=ex−ey=0Substitute y=0fy=ex−e0=0⇒ex−1=0⇒ex=1Taking ′ln⁡ on both sides,ln⁡(ex)=ln⁡(1)x=0Thus (0, 0) is the crirtical point.Step 2D(x, y)=f×(x, y)fyy(x, y)−(fxy(x, y))21) If D(a, b)&amp;gt;0 amd f×(a, b)&amp;gt;0 then (a, b) is local minimum of f2) If D(a, b)&amp;gt;0 and f×(a, b)&amp;lt;0 then (a, b) is local maximum of f3) If D(a, b)&amp;lt;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=exD(x, y)=yex(−ey)−(ex)2=−yex+y−e2xThe critical point is (0, 0)

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

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