Question

How do you use the Intermediate Value Theorem to show that the polynomial function `f(x) = x^3 -3x^2 + 3` has one zero?

Answer 1

See the explanation.

Explanation:

To show that the function has at least one zero, find a value of `x` for which `f(x)` is positive and another for which `f(x)` is negative.
Because polynomial functions are continuous everywhere, `f` will be continuous on the closed interval from the lesser of the `x` values to the greater.

For example, `f(-10)` is clearly negative and `f(10) = 1000-300+3` is positive.

We write a proof:

`f` is continuous on `[-10,10]`.
`0` is between `f(-10)` and `f(10)`, so, by the ontermediate value theorem,

there is at least one `c` in `[-10,10]` with `f(c) = 0`.

(In fact there are three such `c`'s.)