How do you identify an interval [f,g] so that the Contraction Mapping Theorem guarantees convergence to the positive fixed point for the following:
a) $\frac{14-x^3}{13}\ $ b) $e^{-x}$
I tried drawing a graph and i see it visually but am not so sure about proving it rigorously.
a) Let $f(x) = \frac{14 - x^3}{13}$, then $|f'(x)| < 1 \iff |-3x^2/13| < 1 \iff x \in (-\sqrt{13/3}, \sqrt{13/3}).$
b) Similiarly, let $g(x) = e^{-x}$, then $|g'(x)| < 1 \iff |e^{-x}| < 1 \iff x \in (0, \infty).$