**Beverly asked:**

Looks like you have a simplification error in the second derivative. You have:

**Dr. Callahan Answered:**

**Beverly asked:**

Looks like you have a simplification error in the second derivative. You have:

f”(x)=12x^2-4=4(x-1)

But it should be

f”(x)=12x^2-4=4(3x^2-1)

then the intervals for concavity all change.

Am I missing something? Thank you so much for your good product and wonderful support!

You are correct. Thanks for pointing that out.

See the plot of the original equation (a good idea is to look at plots when things are not working out)

From here we can see how it is shaped.

Now if we look at the second derivative, it should be

and look like

which has roots

and looks like

The concavity chart will now show up, down, and up but around the points

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