Question

find all points where the function has any relative extrema or saddle points and identify the type of relative extremum. f(x,y)=12xy
a. saddle point at (0,0)
b. relative maximum at (0,0)
c. relative minimum at (-1,-1), saddle point at (0,0)
d. no relative extrema or saddle points

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 \((0,0)\)