contestada

A continuous function y=f(x) is known to be negative at x=0 and positive at x=1. Why does the equation f(x)=0 have at least one solution between x=0 and x=1?

Respuesta :

bcalle
Because the y values change signs from negative to positive which means the graph had to cross over the x-axis thereby creating a solution.

Answer:

Givens

It's known that the function is negative at [tex]x=0[/tex] and positive at [tex]x=1[/tex].

To answer this question, we need to use the Intermediate Value Theorem, which states:

  • If f(x) is a continuous function on [a, b], then for every k between f(a) and f(b), there exists a value c belongs to (a, b) such that f(c) = k.

Therefore, the function [tex]f(x)=0[/tex] hast a least one solution between [tex]x=0[/tex] and [tex]x=1[/tex] because the function is continuous in the closed interval [0,1], and based on the intermediate value theorem, if there exist a value inside this interval, then there's at least one solution there.

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