Proof for positivity? (1 Viewer)

[Lukybear]

Show that sinx - x + x^3/6 >=0 for x>0

 

[adomad]

well u only need to prove it for 0<x<1 because, any number bigger than 1 will make the eqn positive as x^3 dominates...i may be wrong... im thinking of tangents to sinx curve to prove it but i am going no where

 

[Lukybear]

Well Q says to prove using f'(x)>0 for x>0 and f(0) = 0

But f'(x) = x^2/3 +cosx-1
So mabey im thinking of proving x^2/3+cosx >1 for x>0
But how do i do that? Was thinking about graphing and showing it. But i think we need someform of higher formalised proof.

 

[gurmies]

 

[Lukybear]

O thxs. Thats very helpful. Would of never thought of that.

 

[gurmies]

No worries. http://pages.pacificcoast.net/~cazelais/187/maclaurin-sin.pdf should provide some intuition as to where they derived the problem from.