halcyon cedar
How do i solve this, i clearly cannot use the mean value theorem for tan near pi/2, and i also am not sure of taylor expansion either since i dont know when approximations are reliable and when they aren't, especially since 0,pi/2 looks like a very large interval to rely on such error filled approximations do i use mean value theorem on 0,pi/4 and start working on pi/4 to pi/2 separately?
Or does this all have a different way to solve? I feel like im doing this wrong 
halcyon cedar
How do i solve this, i clearly cannot use the mean value theorem for tan near pi...
The sh function is the hyperbolic sine function sinh
Actually I was able to use the mean value theorem for 0 to X, but now I'm struggling to show that the derivative that I found is positive if I managed to find that I can easily deduce that the function is indeed positive
for f(x) = 2tanx -shx
f'(c)=f(x)/x exists for all x between 0,pi/2
i want to prove that f'(c) is positive
f'(c) = 2(tan²c + 1) - ch(c)
= [ 2 - cos²c • ch(c) ] /cos²c
where 0 < c < x < pi/2
halcyon cedar
i believe i found the solution, even tho it includes using a calculator to calculate values for cosh values at 0, pi/4, pi/2
Studying each of the two halves of the interval either between 0 and pi/4 or pi/4 and pi/2, has led to the conclusion that f'(c) must be positive.
(i bounded ch(x) and tan(x) between specific values since they are both increasing strictly on each interval)
thus f(x) = xf'(c) >0
==> f(x) > 0
==> 2tanx - shx > 0 for all 0<x<pi/2
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