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)=yex−ey

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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)=yex−ey

joshyoung05M · Accepted Answer

To find the critical point, find fx,fy and set fx=0,fy=0 fx=tex=0⇒y=0 or ex=0 For ex=0 have no solution for x∈R Therefore y=0 Set fy=0 fy=ex−ey=0 Substitute y=0,fy=ex−e0=0⇒ex−1=0⇒ex=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)&amp;gt;0 and f×(a,b)&amp;gt;0 then (a,b) is local aximum of f 2) If D(a,b)&amp;gt;0 and f×(a,b)&amp;lt;0 then (a,b) is local aximum of f 3) If D(a,b)&amp;lt;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+y−e2x The critical point is(0,0) D(0,0)=−0e0−e2⋅0=−e0=−1 Thus D(0,0)=-1&amp;lt;0 then f(0,0) is saddle point, f(0,0)=(0)e0−e0=−e0=−1 Therefore -1 is saddle point.

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

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