Then according to your diagram, extend an intersection boundary and point L.

Then obviously | AL | >；; |AN|，AL*AM=|AL|*|AM|*cos∠MAL

& gt|AN|*|AM|*cos∠MAN

That is, when AM*AN takes the maximum value, n must be at the boundary point!

1. When in the advertisement

Obviously, point D is the largest, and it also belongs to DC, which is included in the discussion of DC side.

2. when I was in DC

A (0,0)

M(2， 1)

N(x，2)0 & lt； = x & lt=2

AM=(2， 1)

AN=(x，2)

AM*AN = 2x + 2

Obviously, when x=2, that is, when n reaches point C, it reaches the maximum, that is, 6.

3. When on AB

Obviously, point B is the largest, and B also belongs to BC and is included in the discussion of BC.

4. When I was in BC

A (0,0)

M(2， 1)

N(2，y)0 & lt； = x & lt=2

AM*AN = 4 + y

Obviously, when y=2, that is, n reaches its maximum value at point C, that is, 6.

To sum up, know

When n reaches point c,

AM*AN = 6, maximum.