Sign charts (extrema & inflection)

We use sign charts to pinpoint important x-values on a given function. They are especially useful when we're not given a visual of the function in a graph.

For example, if I gave you the following function...

...and asked you to pinpoint the x-values for any peaks / valleys (local extrema) or points at which the concavity changes (inflection points), we could use a sign chart to achieve this!

Let's start with creating a sign chart depicting the extrema in this function (with f'(x)). Then, we'll move onto creating a sign chart depicting the inflection points for this function (with f''(x))

Sign chart for extrema (f'(x))

To create our sign chart for extrema, we must first find f'(x)!

Now, we must set f'(x) to zero, and find the corresponding x-values.

Remember, at the apex of peaks and valleys (a.k.a. local extrema) in our function, the slope is zero...

...because at that point, the slope is a horizontal line!

This results in x = ±1!

Let's place these x-values on our sign chart.

Now, we must plug in x-values less than -1...

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