# Concept derivation guides

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Concept guide



## [Bisection: four bracket updates, the invariant and the separate proof obligation for root existence](/en/derivations/derivation-mko-bisection-method-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-bisection-method)

For f(x)=x²−2, start from [1,2] with the supplied endpoint signs f(1)<0 and f(2)>0, run four bisection steps with the supplied midpoint signs and trace the brackets; prove that 'opposite endpoint signs and a<b' is an invariant, that the width after n steps is 1/2^n, and that, in an extended run with every needed sign available and effective updates, a width-based stopping rule terminates after finitely many steps; then show that the bracket data alone do not prove a root exists, and separately cite the intermediate value theorem to conclude, for the continuous f, that the final bracket contains a root.

Finite sign-bracket updates in the bisection method

derivation-mko-bisection-method-01





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