Analyzing critical points 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)=λ1+x2+y2

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Analyzing critical points 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)=λ1+x2+y2

Bucharly9 · Accepted Answer

Maxima or minima:  If f(x, y) is any function.  1) Evaluate fx=0 and fy=0 for critical point.  2) Evaluate D=(f×)(fyy)−(fxy)2 at critical point,  If D&amp;gt;0 and  f×&amp;gt;0  , function is minima at critical point.  If D&amp;gt;0 and  f×&amp;lt;0  , function is minima at critical point.  If D&amp;lt;0 , function saddle point.  If D=0, cant

Addison Matthews · Accepted Answer

Now again partial differentiate fx with respect to x, and fy with respect to x and y,  f×(x,y)=∂∂x(1−x2+y2(1+x2+y2)2)  =(1+x2+y2)2∂∂x(1−x2+y2)−(1−x2+y2)∂∂x(1+x2+y2)2(1+x2+y2)4  =(1+x2+y2)2(−2x)−(1−x2+y2)2(1+x2+y2)(2x)(1+x2+y2)4  =(1+x2+y2)(−2x)−(1−x2+y2)2(2x)(1+x2+y2)3  =−2x−2x3−2xy2−4x+4x3−4xy2(1+x2+y2)3  =−6x+2x3−6xy2(1+x2+y2)3  fyy(x,y)=∂∂x(−2xy(1+x2+y2)2)

Analyzing critical points Find the critical points of the following

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