From what you have described, it would appear that your triangle has vertices:

and I assume that .

Therefore its base is and its height is .

So its area is given by the formula:

.

Since you want the maximum area:

.

Setting the derivative equal to and solving for :

or .

But since for all , that means that only is valid.

Therefore .

To check that this IS a maximum, find the second derivative and check that it is negative at .

At

.

This function is negative at , so does indeed maximise the area.

So at :

square units.