Use Newton's Method to find the smallest positive value of a so that the graph of...?

Question

...f⁡(x) = (x)⁢sin⁡(4⁢x) has a horizontal tangent line at x=a. Answers correct to three decimal places.

Answer 1

f' = sin(4x) + 4xcos(4x) = 0
tan(4x) + 4x = 0
We are looking x>0, so tan(4x) < 0. Pi/2 < 4x Pi/8 < x < Pi/4
x' = x - F(x)/F'(x) = x - (tan(4x) + 4x)/(4 + 4/cos^2(4x))
Now you have to start calculations.

Answer 2

Newtons method is an iteration algorithm for finding the root of a function F - a root is a place where the value of the function is 0.

You start with a guess, x, and compute a new approximation, x' with the formula:

x' = x - F(x)/F'(x)

where F'(x) is the derivative of the function F at x. If the result is good enough (i.e. if F(x) is close enough to 0), you are done. Otherwise, you iterate again using x' as your guess, and again, etc. as needed.

Newton's method is only guaranteed to work if the function is smooth, there is a simple root, and the starting guess is close enough to the root. And when it works, it gives you better and better approximations to the "true" answer, but it almost never actually gives the exact true answer.

So what you need are:

a) a function F(x)
b) a good enough starting guess
c) a stopping tolerance (how accurate do you have to be?)

Since we are looking for a place where the tangent is horizontal, we are looking for a root of the derivative.

But not just any root. x = 0 is a root of the derivative but not the one we want. Therefore our starting value has to be far enough away from 0 so that Newton's method find the smallest positive value and doesn't just go to 0.

There are a number of ways to come up with a good guess.

On simple approach is to look at the function and note that both x and sin(4x) keep rising until 4x = pi/2. At that point sin(4x) starts falling even though x keeps rising.

By the time 4x = pi, the value of the function has dropped to 0 again.

The derivative is 0 when the function is at maximum or minimum, so it looks like we want to find the maximum that is between 4x = pi/2 (i.e. x = pi/8) and 4x = pi (i.e. x = pi/8).

So how about starting at pi/6?