find all points where the function has any relative extrema or saddle points and identify the type of…

find all points where the function has any relative extrema or saddle points and identify the type of relative extremum. f(x,y)=12xy\na. saddle point at (0,0)\nb. relative maximum at (0,0)\nc. relative minimum at (-1,-1), saddle point at (0,0)\nd. no relative extrema or saddle points

find all points where the function has any relative extrema or saddle points and identify the type of relative extremum. f(x,y)=12xy\na. saddle point at (0,0)\nb. relative maximum at (0,0)\nc. relative minimum at (-1,-1), saddle point at (0,0)\nd. no relative extrema or saddle points

Answer

Explanation:

Step1: Find critical points

Find first - order partial derivatives. For (f(x,y)=12xy), (f_x = 12y) and (f_y=12x). Set (f_x = 0) and (f_y = 0). (12y = 0) gives (y = 0), (12x=0) gives (x = 0). So the critical point is ((0,0)).

Step2: Find second - order partial derivatives

(f_{xx}=0), (f_{yy}=0), (f_{xy}=12).

Step3: Use the second - derivative test

The discriminant (D=f_{xx}f_{yy}-(f_{xy})^2). Substitute the values: (D=(0\times0)-12^{2}=- 144<0).

Answer:

A. Saddle point at ((0,0))