Definition: Let f(x) be a differentiable function, and let (a,f(a)) be a critical point of f(x). Then,
1. If f”(a)>0, then (a,f(a)) is a local minimum of f(x).
2. If f”(a)<0, then (a,f(a)) is a local maximum of f(x).
3. If f”(a)=0, then the second derivative test tells us nothing!
The second derivative test has strengths and weaknesses. It is certainly quicker than the first derivative test; but it fails to tell us anything at certain points. Let’s see it in action.
Example: Use the second derivative test to classify the critical points of f(x) = 3x5 – 5x3.
Solution: From the example in First Derivative Test page, we find the critical points to be (−1,2), (0,0), and (1,−2).
We take the second derivative of the function by taking the derivative of 15x4 – 15x2:
f”(x) =60x3 – 30x.
Again, we look at each point individually:
f”(−1) = -30, so (−1,2) is a local maximum since f”(−1)<0.
f”(0)=0, so the second derivative test fails to work.
f”(1)=30, so (1,−2) is a local minimum since f”(1)>0.
As you can see, the second derivative is much faster than the first derivative test, but it doesn’t always work. Be sure to know both tests, so that you can classify all critical points as fast as possible!
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