Question

How do you prove that `f(x) = x^5 - 2x^3 - 2` has a root in the interval `[0, 2]`?

Answer 1

Use the intermediate value theorem

Explanation:

Letting `f(x) = x^5 - 2x^3 -2`.

We see that

`f(0) = 0^5 - 2(0)^3 - 2 = -2`

and

`f(2) = 2^5 - 2(2^3) - 2 = 14`

It's clear that `f(0)` is negative and `f(2)` is positive. Therefore, somewhere on `[0, 2]`, `f(x)` must cross the x-axis, so there must be at least one root in that interval.

Hopefully this helps!