Range of x[0] for Newotn Raphson method to be valid

[Harmony]

1. The problem statement, all variables and given/known data
Using Newton-Raphson's method, find the solution to the equation x = tan x in the interval \pi/2 to 3\pi/2. Find an interval, in which any starting value generates a sequence that converges to the solution .


2. Relevant equations
Newton Raphson's Method


3. The attempt at a solution
By trial and error method and the aid of MAPLE software, I found the interval to be [4.28765790535, 4.71238871734]

Is it possible to solve the question without using trial and error method? The newton raphson equation is a sequence, so in my opinion, the sequence is only convergent for certain range of x[0]. The range require by the question would be the range fall within the interval of the solution.

Is this method feasible?
Can anyone give me hints or external reference to solve this question?

[CompuChip]

Well, you could try manually doing some steps for different starting points and see if you can find out where it goes wrong. For example, I can imagine that when you start at a point where the graph is nearly flat, you will get a quick divergence. Try to capture your idea in a formula and then see if you can get the boundary.

[bigevil]

I suggest you draw a graph of tan x and x and look at where the intercept is. Realistically, you can only expect the intercept to be between π and 3π/2. The question helpfully limits the integers for you to choose from already.

If you intend to explore, find f'(x) = \frac{d}{dx} (x - tan x) and make sure that for the integer you choose f'(x) does not equal 0 or approach 0 too closely. This is because near turning points or inflexion points the function changes its concavity too much for you to do any useful analysis there.