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

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

For instance, consider the following graph of

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

Hopefully this helps!

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Please see below.

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

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?

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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.

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