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

find all points where the function has any relative extrema or saddle points and identity the type of relative extrema\nf(x,y) = 7xy\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 identity the type of relative extrema\nf(x,y) = 7xy\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 first - order partial derivatives

The first - order partial derivatives of (f(x,y)=7xy) are: (f_x=\frac{\partial f}{\partial x}=7y), (f_y=\frac{\partial f}{\partial y}=7x)

Step2: Find critical points

Set (f_x = 0) and (f_y=0). From (f_x = 7y = 0), we get (y = 0). From (f_y=7x = 0), we get (x = 0). So the critical point is ((0,0))

Step3: Find second - order partial derivatives

(f_{xx}=\frac{\partial^2 f}{\partial x^2}=0), (f_{yy}=\frac{\partial^2 f}{\partial y^2}=0), (f_{xy}=\frac{\partial^2 f}{\partial x\partial y}=7)

Step4: Use the second - derivative test

The discriminant (D=f_{xx}f_{yy}-(f_{xy})^2) Substitute (f_{xx} = 0), (f_{yy}=0), (f_{xy}=7) into the formula: (D=(0\times0)-7^2=- 49<0)

Answer:

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