Question

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]?

Answer 1

`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.]