Finding Area of a Right Triangle & Proving arctanx Limit

[azatkgz]

Question1:In a right triangle,the sum of the lengths of the hypotenuse and a leg is 1.Find the largest possible area of such triangle.
Question2:Prove that arctanx=pi/2-1/x+o(1/x)(x tends to +inf)
Question3:evaluate lim(2arctanx/pi)^2 (x tends to +inf)

 

[sutupidmath]

well for the first one here it is my approach, i hope i am not wrong.
let c be the hypotenuse, and a, b to legs of that right triangle. Let us take
c+b=1 ----------*
we know that the area of a right triangle is

A=(ab)/2, from * we have A=(a/2)(1-c),, so this is a function, and we need to find its maximum. finding the first derivative we get A'=1-c-a, so the maximum of this function is when a=1-c, so the maximum area is A=(b^2)/2, or (1-c)^2/2

the tird one, i guess the limit is 2pi, i did not do any calculations though, just glanced it.
and for the second one, try to expand arctanx using the taylor formula, when n=2, i mean just applying the derivative twice.
i hope i was helpful.