Find all points where the function has any relative extrema. Identify any saddle points.
\[
f(x, y)=3 x^{2}-2 x y+y^{2}-16 x+8
\]
Find the derivatives $f_{x x}, f_{y y}$, and $f_{x y}$.

Step 1 ：First, we need to find the first order partial derivatives of the function. The partial derivative of \(f(x, y)\) with respect to \(x\) is \(f_x = 6x - 2y - 16\), and the partial derivative of \(f(x, y)\) with respect to \(y\) is \(f_y = -2x + 2y\).

Step 2 ：Next, we set these partial derivatives equal to zero to find the critical points. Solving the system of equations \(f_x = 0\) and \(f_y = 0\), we get \(x = 4\) and \(y = 2\).

Step 3 ：Now, we need to find the second order partial derivatives of the function. The second partial derivative of \(f(x, y)\) with respect to \(x\) is \(f_{xx} = 6\), the second partial derivative of \(f(x, y)\) with respect to \(y\) is \(f_{yy} = 2\), and the mixed partial derivative of \(f(x, y)\) with respect to \(x\) and \(y\) is \(f_{xy} = -2\).

Step 4 ：Then, we calculate the determinant of the Hessian matrix, which is \(D = f_{xx}f_{yy} - (f_{xy})^2 = 6*2 - (-2)^2 = 8\).

Step 5 ：Since \(D > 0\) and \(f_{xx} > 0\), the function \(f(x, y)\) has a relative minimum at the point \((4, 2)\).

Step 6 ：Finally, we find the value of the function at this point, \(f(4, 2) = 3*(4)^2 - 2*4*2 + (2)^2 - 16*4 + 8 = -16\).

Step 7 ：So, the function \(f(x, y)\) has a relative minimum of \(-16\) at the point \((4, 2)\), and there are no saddle points.