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Consider the function $\displaystyle{ f(x,y) = (4 x - x^2)(12 y - y^2)}$ .

Find the first- and second-order partial derivatives of $f$ .

$f_x =$
$f_y =$
$f_{xx} =$
$f_{xy} =$
$f_{yy} =$

Find and classify all critical points $(x,y)$ of the function. If there are more blanks than critical points, leave the remaining entries blank.

There are several critical points to be listed. List them lexicographically, that is in ascending order by $x$ -coordinates, and for equal $x$ -coordinates in ascending order by $y$ -coordinates (e.g., (1,1),(1,10), (2, -1), (2, 3) is a correct order)

In lexicographic order:

The critical point with the smallest $x$ and $y$ coordinates is . Classification:
The next critical point is . Classification:
The next critical point is . Classification:
The next critical point is . Classification:
The next critical point is . Classification:

You can earn partial credit on this problem.