Question

Is it possible to determine the critical points of a function without using the function's derivatives?

Answer 1

Technically yes, if you're given the graph of the function.

For instance, consider the following graph of `y = x^2 - 1`.

graph{y = x^2 - 1 [-10, 10, -5, 5]}

When looking for critical numbers, we will either have a horizontal tangent or a vertical tangent. Here we can draw a horizontal tangent at `x = 0`, therefore, this is a critical number.

Hopefully this helps!

Answer 2

Please see below.

Explanation:

I think it depends on what you mean by "determine" and "using the function's derivative".

A critical number for a function is a number in the domain of the function where the derivative is `0` or fails to exist.

If you look at the graph and think about tangent lines and their slopes, (like where they are horizontal and where they fail to exist) then are you "using the derivative"?

If that does not count as "using the derivative", then it is possible to approximate critical numbers from a graph.

Is approximating the same as determining?