Proving two problems : Derivative roots & trigonometric equetion

[Asaff4]

Hey guys
I was doing my homework and got stuck with these two hard (For me at least)
problems.
I really hope you could help me with those.
Thanks a lot for your time :-D

Problem number one
Prove that between every two roots of the equation \(\displaystyle arctan\left ( x \right ) = 2sin\left ( x \right )\) there is a root of the equation \(\displaystyle 2(1+x^{2})cos\left ( x \right ) =1\).

Problem number two
\(\displaystyle f(x)\) can be differentiated twice in R, show that if \(\displaystyle f(x)\) has \(\displaystyle k\) different roots \(\displaystyle (k\geq 3)\) then \(\displaystyle f'(x)\) has at least \(\displaystyle k-1\) different roots and \(\displaystyle f''(x)\) has \(\displaystyle k-2\) different roots.

 

[Steven G]

Hey guys
I was doing my homework and got stuck with these two hard (For me at least)
problems.
I really hope you could help me with those.
Thanks a lot for your time :-D

Problem number one
Prove that between every two roots of the equation \(\displaystyle arctan\left ( x \right ) = 2sin\left ( x \right )\) there is a root of the equation \(\displaystyle 2(1+x^{2})cos\left ( x \right ) =1\).

Problem number two
\(\displaystyle f(x)\) can be differentiated twice in R, show that if \(\displaystyle f(x)\) has \(\displaystyle k\) different roots \(\displaystyle (k\geq 3)\) then \(\displaystyle f'(x)\) has at least \(\displaystyle k-1\) different roots and \(\displaystyle f''(x)\) has \(\displaystyle k-2\) different roots.

The rules of this forum (have you read them?) states that you need to show your attempt at the problems you post.
For problem 1, have you found the roots of \(\displaystyle arctan\left ( x \right ) = 2sin\left ( x \right )\) ? Try using the double angle formula.

 

[Ishuda]

Hey guys
I was doing my homework and got stuck with these two hard (For me at least)
problems.
I really hope you could help me with those.
Thanks a lot for your time :-D

Problem number one
Prove that between every two roots of the equation \(\displaystyle arctan\left ( x \right ) = 2sin\left ( x \right )\) there is a root of the equation \(\displaystyle 2(1+x^{2})cos\left ( x \right ) =1\).

Problem number two
\(\displaystyle f(x)\) can be differentiated twice in R, show that if \(\displaystyle f(x)\) has \(\displaystyle k\) different roots \(\displaystyle (k\geq 3)\) then \(\displaystyle f'(x)\) has at least \(\displaystyle k-1\) different roots and \(\displaystyle f''(x)\) has \(\displaystyle k-2\) different roots.

As an example of the way I would go the about the problem I'll play around with k=2 [you should be more rigorous about it]: First assume we have
f(x_j) = 0, j = 1, 2; x₁ \(\displaystyle \lt \) x₂.
What does the Mean Value Theorem say about the existence of zero derivative between x₁ and x₂. What would happen if x₂ < x₁? What if x₂ = x₁? Now generalize to k zeros, k>2.

 

[Asaff4]

For the first problem, how can I find the roots of such equation 0.0? I thought of some identities but couldn't manage to solve it...
For the second problem, I got it thanks to your help! Thanks a lot!