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Calculus: The Second Derivative Test


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About this Lesson

  • Type: Video Tutorial
  • Length: 3:27
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 37 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus Review (48 lessons, $95.04)
Calculus: Curve Sketching (20 lessons, $25.74)
Calculus: Graphing Using the Derivative (4 lessons, $5.94)

Where the first derivative test is a method used to determine whether critical points are maxima or minima, the second derivative test is used to test concavity for a critical point. If the derivative of a function at a critical point equals zero and the second derivative of that function at that point is positive, you have a minimum (min) and a concave up curve. If the derivative of a function at a critical point equals zero and the second derivative at that same point is negative, you have a maximum (or max) in a curve that is concave down. If the first derivative is zero and the second derivative is zero, the second derivative test is inconclusive - it doesn't mean that the point isn't a max or a min. It just means you need to go back to the first derivative test.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hôpital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Recent Reviews

Calculus: The Second Derivative Test
~ Ferddy


Calculus: The Second Derivative Test
~ Ferddy


Curve Sketching
Graphing Using the Derivative
The Second Derivative Test Page [1 of 1]
All right. Before moving on to looking at functions that actually have denominators in them, I thought I would just mention really quickly something that you may have seen either in class or in your book or in discussions with friends, and that is the second derivative test. Now, we already saw the first derivative test. That’s the means of determining whether critical points are really max or mins by looking about what the derivative is doing before and after. If the derivative is rising—positive—and then falling—negative—then you know max. And if the derivative is negative and then positive, you know min. That’s the first derivative test.
Now, there’s a second derivative test, which is actually quite nice in a lot of applications. Let me just tell you about that really fast. Suppose that we had a min like this, okay? Well, if we have a min like this, what do you know about the second derivative? Well, looking at this, then, you can see this is concave up.
So, in fact, if you find the place where the derivative is zero and the second derivative there is positive, what would that mean? Well, it means that the derivative is zero, so the curve is leveling off, but since the second derivative is positive, you know it has to be concave up somehow. Well, how can you be concave up and have a zero slope? You must be at the very bottom of this thing. So, therefore, it’s a min.
So, in fact, the second derivative test says, if you find a candidate for a max or min, you can just plug that candidate—you don’t have to look to the right and to the left—just plug that candidate into the second derivative. And if you get something that’s positive, you know it must be a min, and why? Because it’s concave up but level there.
What about if we have a max? Well, similar idea. If you have a max and a place where the derivative is zero—so you have a tangent line that’s horizontal—how could that be? Well, you must be, in fact, going up and then down. So, therefore, we must have a second derivative that’s negative, because it’s concave down.
So, if you have a candidate for max/min, in particular, a place where the derivative equals zero, and you plug that number in—don’t look to the right and the left—just plug that number into the second derivative. If you get something that’s negative, then you know it’s got to be concave down but level there. If you’re concave down and level, the only place here where you’re concave down and level is on the top. Therefore, it’s a max.
So, the second derivative test says if you have the place where the derivative equals zero and the second derivative at that point is negative, then you have a max. If the second derivative at that point is positive, then you must have a min. Okay. That’s the second derivative test.
What if the second derivative at that point is zero? The answer is the test fails. It doesn’t mean it’s not a max. It doesn’t mean it’s not a min. It means that that test did not work. It was inconclusive. You’ve got to go back to the first derivative test.
So, the second derivative test is something that’s actually really useful and handy if you’re doing some of those max/min word problems. Instead of looking to the right and the left, you might want to just take a second derivative. However, when you’re graphing functions, I don’t think it’s a great idea. First of all, it doesn’t work for those little sharp points like this. Only if you’re looking at points where the derivative equals zero. And secondly, you’ve got to look to the right and left to see if the function’s increasing or decreasing anyway, so I don’t see it really saving that much time.
But anyway, it’s a good idea. It’s a very pretty idea—the notion that just by looking at the second derivative out of place where the derivative is zero, we can see if it’s a max or min. Second derivative test. Enjoy.

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