Suppose you have $f(x)=e^{-2x}-1=0$ then an iterative form of this equation would be $x_{n+1} = g( x_n )=\ln(2 x_n +1)$, by the IVT I know $f(x)$ has a root t in the interval $[1,2]$ and the fixed point of $g(x)$ between $1$ & $2$ is $t$. Hence, how do I show that the iterative form of $f(x)$ converges to this to this $t$ by showing that $g$ is a contraction on $[1,2]$ and how do I find the Lipschitz constant to verify this?
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how do I find the Lipschitz constant --- There are (uncountably) infinitely many Lipschitz constants whenever a Lipschitz constant exists, and if by "the Lipschitz constant" you mean the smallest such constant, then (assuming I'm not overlooking something) you don't need the smallest such constant to verify that $g$ is a contraction (you only need a Lipschitz constant that is less than $1).$ – Dave L. Renfro Mar 22 '20 at 17:04
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Please check your equations, the equation for $f$ gives $x=0$ as only solution, which is different from the fixed-point equation which has an additional solution. – Lutz Lehmann Mar 22 '20 at 20:57