From introduction to analysis,by Arthur P. Mattuck,problem 20-1.
I am stuck in the sub-problem (d) of this problem,especially the magic number 2.5,please help,thanks.
One way of rigorously defining the trigonometric functions is to start with the definition of the arctangent function. (This is the route used for example in the classic text Pure Mathematics by G. H. Hardy.)
So, assume amnesia has wiped out the trigonometric functions (but the rest of your knowledge of analysis is intact). Define
(a) Prove T(x) is defined for all x and odd.
(b) Prove T(x) is continuous and differentiable, and find T(x).
(c) Prove T(x) is strictly increasing for all x; find where it is convex, where
concave, and its points of inflection.
(d) Show T(x) is bounded for all x, and |T(x)| < 2.5, using comparison
of integrals. Can you get a better bound?