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

Question

Emma Aldridge · Accepted Answer

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!

Luke Alvarez · Answer

Please see below.

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?

Charles Arocha · Answer

undefined No, it is not possible to determine the critical points of a function without using its derivatives. Critical points are found by setting the derivative of the function equal to zero and solving for the corresponding x-values. These critical points can be maxima, minima, or points of inflection on the graph of the function. Therefore, derivative information is necessary to identify critical points.