Geometry of Functions VIII: The Second Derivative Test
Primary tabs
SAMPLE LESSON
Sign in for easy to read lessons narrowly focused on a specific subject. All lessons are linked to each other allowing you to easily see the connections between different Calculus concepts. Our unique Calcmap will give you a bird's-eye view of all our lessons and help you see how important concepts are related to each other. All of the formulas are available to our subscribers, but here is a sample of the lesson you will see when you log in:
The Second Derivative Test is a method for determining if a critical point is a maximum or a minimum. It requires that a function be twice differentiable at the critical point. If you have been following along with our Geometry of Functions Series then the Second Derivative Test should be a straight forward extension of concepts you already know.
Let's motivate the Second Derivative Test by examining a maximum. We know from the First Derivative Test that for a maximum the slope of the function is positive on the left of the critical point and negative on the right of the critical point. So as we go from left to right, the slope goes from being positive to negative. If we denote the slope by $m$ then we know that
<Sign in to see all the formulas>
as we cross the critical point.
Using Leibniz Notation we know that
<Sign in to see all the formulas>
But since the slope $m$ is the first derivative, <Sign in to see all the formulas>, we know that
<Sign in to see all the formulas>
and therefore
<Sign in to see all the formulas>
A similar argument hold for a minimum:
Ok, now that we have motivated the Second Derivative Test we need to actually state what it is:
| The Second Derivative Test |
|---|
Let's assume that $e$ is a critical point of $f(x)$ and that <Sign in to see all the formulas>.
|
The proof of the Second Derivative test for a maximum may be found here and the proof for a minimum may be found here.
Finally, it is important to note that the Second Derivative Test is not as flexible as the First Derivative Test. This is because the function has to be twice differentiable at the critical point and if it is twice differentiable then the second derivative must not be zero.
