Second Derivatives and Beyond
Second Derivative Test
- Assume f is defined and twice differentiable on the whole real line. Around a minimum of the function f, is f concave up or concave down?
- Assume f is defined and twice differentiable on the whole real line. Around a maximum of the function f, is f concave up or concave down?
A minimum of f will usually occur at the bottom of a right-side up bowl:
Having a right-side up bowl means f is concave up here.
A maximum of f will usually occur at the top of an upside-down bowl:
Having an upside-down bowl means f is concave down here.
The Second Derivative Test says
- If f is concave up around a critical point, that critical point is a minimum.
- If f is concave down around a critical point, that critical point is a maximum.
This is true because if f is concave up around a critical point, f looks like this:
Such a critical point must be a minimum. On the other hand, if f is concave down around a critical point, then f looks like this:
Such a critical point must be a maximum.
Be Careful: If f " is zero at a critical point, we can't use the Second Derivative Test, because we don't know the concavity of f around the critical point.
Be Careful: There's sometimes confusion about this test because people think a concave up function should correspond to a maximum. This is why pictures are useful. If we remember what a concave up function looks like, we'll be fine.
There's a good question that most people have right about now: if you're not told which to use, how do you know whether to use the first derivative test or the second derivative test?
The good news is that it often doesn't matter. When it's possible to use both the first derivative test and the second derivative test, they will give the same answer.
The other good news is that you can usually do whichever test is easier. Sometimes finding the second derivative is not fun, like with the function
The first derivative is
and while we could find the second derivative, it's not pretty and we don't want to bother. In this case, it probably makes more sense to plug in a couple of numbers and see what the sign of the first derivative is doing. Sometimes the second derivative test doesn't work at all (if f " is 0 at the critical point), in which case we need to use the first derivative test.
On the other hand, sometimes you can see that the second derivative is really nice. Take the function
f (x) = x2 + 4x + 1.
The first derivative is
f '(x) = 2x + 4
and the second derivative is
f "(x) = 2,
which is always positive. Therefore f is always concave up, so any critical point needs to be a minimum. The second derivative test for this one is a piece of cake. Mmm, cake.
The bad news is that, as with the rest of math, we do need to practice. The more functions we stare at, the better we will become at deciding whether to use the first derivative test or the second derivative test to classify a function's extreme points. Don't worry; there are plenty of practice problems.