Call x and y the sides of the right triangle.
A constraint for this optimization problem is x2 + y2 = 22 = 4, from the Pythagorean theorem.
Solve for one of the variables in the constraint. y = √(4 - x2)
The objective function (the function we're trying to optimize) is A = 0.5xy
Plug in what was obtained for y
A = 0.5x√(4 - x2)
Take the derivative of both sides with respect to x and set that to 0
dA/dx = 0.5x *2x/(2√(4 - x2)) + 0.5√(4 - x2) = 0
-0.5x2/√(4 - x2) + 0.5√(4 - x2) = 0
-x2/√(4 - x2) + √(4 - x2) = 0
-x2 + (4 - x2) = 0
-2x2+ 4 = 0
x2 = 2
x = √2
Plug this back in to get y = √2
So the maximal area = 1