How do you use the intermediate value theorem to explain why f(x)=x^3+3x-2 has a zero in the interval [0,1]?

1 Answer
Alan P.
Nov 19, 2016

f(x) is continuous in the domain x in [0,1] and
f(0) < 0 < f(1)
therefore EE_c : f(c) = 0 (by the intermediate value theorem)

Explanation:

The intermediate value says:

If f(x) is a continuous function over the domain x in [a,b]
then for any value k in [f(a),f(b)]
there is a value c such that f(c)=k

For the given function f(0] = -2 and f(1)=+2
Since k=0 in [-2,+2]
there is a value c such that f(c)=0

[Technically, I have assumed without proof that f(x) is continuous within the given domain. If you need this proof, add as a new question or post a comment.]

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