Find the critical points of the following functions. f(x, y)=x4+y4−4x−32y+10

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Find the critical points of the following functions.  f(x, y)=x4+y4−4x−32y+10

likvau · Accepted Answer

Step 1  The function given is f(x, y)=x4+y4−4x−32y+10  Take the first derivative of the function,  fx(x, y)=4x3−4  4x3−4=0  4(x3−1)=0  x=1  fy(x, y)=4y3−32  4y3−32=0  4(y3−8)=0  y3−8=0  y=2  Step 2  The critical are: (1, 2)  Take the second derivative of the fx(x, y)  f×(x, y)=12x2  At (1, 2),  f×(1, 2)=12⋅12  =12  Take the second derivative of the fy(x, y)  fyy(x, y)=12y2  Step 3  At (1, 2)  fyy(1, 2)=12⋅22  =48  The other second derivative with respect to y,  fxy(x, y)=0  Consider the condition,  D=f×⋅fyy−(fxy)2  =12⋅48−0  =576  Here, the value of discriminant is greater than 0. Therefore the critical point (1, 2) is a local minimum point.

Find the critical points of the following functions. f(x,\ y)=x^{4}+y^{4}-4x-32y+10

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