Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

sin x

prove that

sin x≥ 2 x/ pi  , x ∈ [0, pi/2]

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: sin x

First you prove this theorem: If f is a differentiable function and a is a point such that f(a) = 0 and f′(a) > 0, there exists δ > 0 such that f(x) > 0 for all x ∈ (a,a+δ).

Proof:

Using the ε–δ definition of f′(a) (setting ε = f′(a)), there exists δ > 0 such that ∀x ∈ (a−δ,a+δ)

i.e.

i.e.

If we let x ∈ (a,a+δ). then x−a > 0 and so f(x) > 0.

Last edited by JaneFairfax (2007-12-07 03:44:00)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: sin x

Now

Then, by the intermediate-value theorem,

By Rolle’s theorem,

By Rolle’s theorem again, this time to f′,

But this is a contradiction as f″(x) = −sin(x) is never 0 for x strictly between 0 and π⁄2.

Therefore it must be that

Last edited by JaneFairfax (2007-12-07 03:46:28)