Posted on

Calc1 Test 4 / Final Exam #6c

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!
Dr. Callahan Answered:
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.
Figure 1 – Original Equation
Now if we look at the second derivative, it should be
12 x^2 - 4
and look like
which has roots
x = -1/sqrt(3)
x = 1/sqrt(3)
and looks like
Figure 2 – 2nd Derivative
The concavity chart will now show up, down, and up but around the points
x = -1/sqrt(3)
x = 1/sqrt(3)