The point (0,0)is a critical point for the function $f ( x , y ) = x ^ { 4 } + y ^ { 4 }$ .What kind of critical point is it?
A)local minimum
B)saddle point
C)local maximum
D)none of these

Question

Quizplus · Accepted Answer

The function $f(x, y) = x^4 + y^4$ has a critical point at (0,0) because the first partial derivatives $f_x = 4x^3$ and $f_y = 4y^3$ both equal zero at (0,0). To determine the nature of this critical point, we examine the second partial derivatives: $f_{xx} = 12x^2$, $f_{yy} = 12y^2$, and $f_{xy} = f_{yx} = 0$. The determinant of the Hessian matrix at (0,0) is $f_{xx}f_{yy} - (f_{xy})^2 = 12*12 - 0 = 144 > 0$, and since $f_{xx} > 0$, the point (0,0) is a local minimum.

The Point (0,0)is a Critical Point for the Function f(x,y)=x4+y4f ( x , y ) = x ^ { 4 } + y ^ { 4 }f(x,y)=x4+y4

The Point (0,0)is a Critical Point for the Function $f ( x , y ) = x ^ { 4 } + y ^ { 4 }$