Examine the function given below for relative extrema and saddle points. $f ( x , y ) =$ $( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 }$
A)The function has a relative minimum at $( 1,3,0 )$ .
B)The function has a relative maximum at $( 1,3,0 )$
C)The function has a saddle point at $( 1,3,0 )$ .
D)The function has a saddle point at $( 0,1,3 )$ .
E)The function has a saddle point at $( 0,1,3 )$

Question

Examine the function given below for relative extrema and saddle points. $f ( x , y ) =$ $( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 }$
A)The function has a relative minimum at $( 1,3,0 )$ .
B)The function has a relative maximum at $( 1,3,0 )$  
C)The function has a saddle point at $( 1,3,0 )$ .
D)The function has a saddle point at $( 0,1,3 )$ .
E)The function has a saddle point at $( 0,1,3 )$

Quizplus · Accepted Answer

The function $f(x, y) = (x - 1)^2 + (y - 3)^2$ is a sum of two squares, each of which is minimized when its variable equals the constant it is subtracted from. Therefore, the function has a relative minimum at the point where $x = 1$ and $y = 3$, making the coordinates of the minimum $(1, 3)$. The value of the function at this point is $f(1, 3) = 0$, so the complete description of the minimum is at $(1, 3, 0)$. There are no saddle points or relative maxima because the function increases as you move away from this point in any direction.

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 $f ( x , y ) =$