Question

When using the first derivative test to find the critical points of a function, do you always have to include `x=0?`

Answer 1

Yes, if `0` is in the function's domain. Note that if the derivative isn't defined at `0`, that is also a critical point.

Explanation:

That is because `0` can be a critical point of a function. For example, consider the function `f(x) = x^2`.

`f'(x) = 2x`, and the root to the equation `f'(x) = 0 => 2x = 0` is

`x = 0`.

At `0`, our `f` also happens to have an absolute minimum, which wouldn't always be the case.

In fact, many such situations exist, so, unless `0` is not in the functions domain, there is no reason why you wouldn't check what happens there.