Examine the function given below for relative extrema and saddle points. $f ( x , y ) =$ $\frac { x y } { 2 }$
A)The function has a saddle point at $( 0,0,0 )$ .
B)The function has a relative maximum at $( 0,0,0 )$ .
C)The function has a relative minimum at $( 0,0,0 )$ .
D)The function has a saddle point at $( 0,0,2 )$ .
E)The function has a relative maximum at $( 0,0,2 )$ .

Question

Quizplus · Accepted Answer

To determine if the function $f(x, y) = \frac{xy}{2}$ has relative extrema or saddle points, we first find its first and second partial derivatives. The first partial derivatives are $f_x = \frac{y}{2}$ and $f_y = \frac{x}{2}$. Setting these equal to zero, we find that the critical point is at $(0,0)$.The second partial derivatives are $f_{xx} = 0$, $f_{yy} = 0$, and $f_{xy} = f_{yx} = \frac{1}{2}$. The determinant of the Hessian matrix (D) at the critical point is calculated as $D = f_{xx}f_{yy} - (f_{xy})^2 = 0 - \left(\frac{1}{2}\right)^2 = -\frac{1}{4}$.Since $D < 0$, the critical point $(0,0)$ is a saddle point. Therefore, the correct choice is that the function has a saddle point at $(0,0,0)$.

Examine the Function Given Below for Relative Extrema and Saddle f(x,y)=f ( x , y ) =f(x,y)=

Examine the Function Given Below for Relative Extrema and Saddle $f ( x , y ) =$