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

Explanation:

Step1: Find first - order partial derivatives

The first - order partial derivatives of (f(x,y)=5xy) are: (f_x=\frac{\partial f}{\partial x}=5y), (f_y=\frac{\partial f}{\partial y}=5x) Set (f_x = 0) and (f_y = 0). From (f_x=5y = 0), we get (y = 0). From (f_y=5x = 0), we get (x = 0). So the critical point is ((0,0)).

Step2: Find second - order partial derivatives

The second - order partial derivatives are: (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}=5)

Step3: Use the second - derivative test

The discriminant (D=f_{xx}f_{yy}-(f_{xy})^2) Substitute (f_{xx} = 0), (f_{yy}=0), (f_{xy}=5) into the formula for (D): (D=(0\times0)-5^2=- 25<0)

Answer:

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