# Existence of a root

• Sep 8th 2009, 07:00 PM
NonCommAlg
Existence of a root
We have two functions $\displaystyle f,g : [a,b] \longrightarrow \mathbb{R}$ such that $\displaystyle g$ is continuous, $\displaystyle f+g$ is decreasing, $\displaystyle f(a) < 0$ and $\displaystyle f(b) > 0.$ Prove that there exists $\displaystyle a < c < b$ such that $\displaystyle f(c) = 0.$
• Sep 8th 2009, 11:08 PM
CaptainBlack
Quote:

Originally Posted by NonCommAlg
We have two functions $\displaystyle f,g : [a,b] \longrightarrow \mathbb{R}$ such that $\displaystyle g$ is continuous, $\displaystyle f+g$ is decreasing, $\displaystyle f(a) < 0$ and $\displaystyle f(b) > 0.$ Prove that there exists $\displaystyle a < c < b$ such that $\displaystyle f(c) = 0.$

Binary search will converge to a point $\displaystyle c$ such that for all $\displaystyle 0<\varepsilon<\varepsilon_0$ (that is for all $\displaystyle \varepsilon$ small enough):

$\displaystyle f(c-\varepsilon)<0$ and $\displaystyle f(c+\varepsilon)\ge 0$

Thus either $\displaystyle f(c)=0$ or there is an increasing jump discontinuity at $\displaystyle c$. But an increasing jump at c is imposible given that g(x) is continuous and f(x)+g(x) and decreasing at c.

CB