TITLE: Why does the Newton-Raphson method not converge for some functions?
QUESTION [37 upvotes]: $f(x)=2x^2-x^3-2$. This is a cubic type graph as shown.  The real root of this graph is $(-0.839,0)$.
So, the question is to use Newton's approximation method twice to approximate a solution to this $f(x)$.
I use an initial starting value of $x_0=1$.
My first approximation is $x_1=2$, and my second one is $x_2=1.5$.
I seem to not move any closer to the real solution as I keep iterating through the method.
Am I misunderstanding how to use this approximation?  Is the issue that my first guess was too far from the actual solution?

REPLY [2 votes]: The Newton-Raphson method basically asks you to draw the tangent to the function at the point $x_0$, and $x_1$ is the point where that tangent hits the $x$-axis. This obviously works well if the function follows reasonably close to the tangent up to the point where $y = 0$, and not so well otherwise. 
You can see that between $x = 0$ and $x \approx 1.4$ the tangent actually points in the wrong direction, away from the $0$. And where the tangent is parallel or almost to the $x$-axis, close to $x = 0$ and $x = 1.4$, following the tangent will get you far away from the zero. 
For $x_0 < -0.346, \ x_1$ will be closer to the solution than $x_0$, and will be in the same range again, so the sequence will converge towards the solution. For $x_0 > 1.5, \ x_1$ will also be closer to the solution than $x_0$, but you need to get over the area from $-0.346 \to 1.5$.
And $\approx 1.081792$ there is a point where $3$ iterations get you back to the point where you started :-)