Question

the legs of a right triangle have lengths a and b satisfying a + b =10. which values of a and b maximize the area of the triangle?

Answer 1

area = (1/2) a b
b = 10 - a

area = (1/2) a (10 -a)

= (1/2) 10 a - a^2

If we were doing algebra we could find the vertex of this parabola but since you said calculus, we will use calculus
dArea/da = 0 when max or min
dArea/da = (1/2)(10 - 2 a)
2 a = 10
a = 5
b = 5

Answer 2

Is not really okay.