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Given a function f(x), you are asked to show the existence of a zero in a given interval, say [a,b], using the intermediate value theorem.
(1) In order to apply the intermediate value theorem, you must first show that f(x) is continuous on the closed interval [a,b].
(2) To ensure the existence of a zero in the interval, you must show that there exists some `x_1 in [a,b]` such that `x_1<0` , and some `x_2 in [a,b]` such that `x_2>0` .(Note that it does not matter whether `x_1<x_2` or `x_1>x_2` , you just need them to have opposite sign.) ** In most textbooks, you will find that f(a) and f(b) have opposite sign **
(3) If you can do both (1) and (2), then the intermediate value theorem applies and there is at least one zero on the interval.
- The intermediate value theorem states that if f(x) is continuous on a closed interval [a,b], and k is some number such that f(a)<k<f(b), or f(a)>k>f(b), then there exists at least one number `c in [a,b]` such that f(c)=k.
For this problem we find values of f greater than and less than zero, so we let k=0.
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