Since this is a right triangle,
a^2 + b^2 = 12^2 = 144, so b^2 = 144 - a^2, and b = sqrt(144-a^2)
ab a*sqrt(144-a^2)
The area of the triangle is A = ----- = -----------------------
2 2
Since the radicand of the square root function cannot be smaller than zero, the domain for a is [0, 12]; the domain for b is also [0,12]. If a increases, b must decrease, and vice versa.
The area A of the triangle can never be less than 0 nor more than what?
You could use calculus to answer this question, or you could make a table of a, b and Area values and then determine from the table which area A is the max.