Solving equations with derivatives

[daniel_i_l]

Homework Statement

Prove or disprove:
1)If a and b are real constants and 0<a<1 then the equation x - a*cosx = b has only one solution.
2)Between every two solutions to arctanx = sinx there's atleast one solution to 1 - cosx = x^2 cosx

Homework Equations

The Attempt at a Solution

1)True: The limit of x - a*cosx at +infinity is +infinity and at -infinity is
-infinity. Also, the derivative is 1+a*sinx > 0 for all x and so it's an injective continues function. Which means that there's one and only one solution for every b. (one solution because of the mean value theorem and only one cause it's injective)

2)True: The derivative of f(x) = arctanx - sinx is 1/(1+x^2) - cos x =
(1 - cos x - (x^2)cosx)/(1+x^2). So if for x_1 and x_2 f(x)=0 then there's some x_1<x_3<x_2 where f'(x) = 0. And since (1+x^2) =/= 0 then
1 - cos x - (x^2)cosx which means that
1 - cos x = (x^2)cosx

 

[VietDao29]

Yeah, all seems ssssooo good to me. They are both perfect. ^.^ Well done, daniel_i_l. :)

 

[daniel_i_l]

Thanks for your reply.