I'm trying to do optional exercises for my numerical methods class. I'm stuck in this one right now:
Consider the function $f(x)=-e^{-2x}+3x$.
a) Prove that $f$ has an unique real root.
b) Prove that the iterative method given by the function $g_1(x)=e^{-2x}-2x$ is not useful to find this root.
c) Prove that the iterative method given by the function $g_2(x)=e^{-2x}/3$ converges to the root for every initial value in $[0,1]$.
d) Give another method "numerically better" than the above one and justify it.
I've solved without problems the a) part, there's a root and it's unique. Then, when I arrived at the part b) and c), I got stuck. Which iterative method they want me to use?
There's more than three (Newton, bisection, fixed point...) to choose, how I'm supposed to know which one they want me to use here? I thought about using the fixed point one, but I'm not sure if this is correct.
Another question that came to my mind is: how we show without initial value that a fixed point method doesn't work?
We've just started the "zeros of functions" section, so any help or explanation would be really appreciated.
Thanks for your time!