Find all the critical points of $f ( x , y ) = x ^ { 3 } - 3 x + y ^ { 2 } - 4 y$ and classify each as maximum, minimum, or saddle point. Select all possible choices.
A)The point (1, 2)is a relative minimum.
B)The point (-1, -2)is a saddle point.
C)The point (-1, 2)is a saddle point.
D)The point (1, 2)is a relative maximum.
E)The point (1, -2)is a relative maximum.

Question

Quizplus · Accepted Answer

The critical points are found by solving the system of equations $f_x = 0$ and $f_y = 0$. This gives the points $(1, 2)$ and $(-1, 2)$. To classify each point, we compute the second partial derivatives:$$f_{xx} = 6x, \quad f_{xy} = 0, \quad f_{yy} = 2$$At $(1, 2)$, we have $f_{xx} = 6 > 0$ and $f_{yy} = 2 > 0$, so this point is a relative minimum.At $(-1, 2)$, we have $f_{xx} = -6 < 0$ and $f_{yy} = 2 > 0$, so this point is a saddle point.

Find All the Critical Points Of f(x,y)=x3−3x+y2−4yf ( x , y ) = x ^ { 3 } - 3 x + y ^ { 2 } - 4 yf(x,y)=x3−3x+y2−4y

Find All the Critical Points Of $f ( x , y ) = x ^ { 3 } - 3 x + y ^ { 2 } - 4 y$