Notes on Textbook Section 4.1
Critical Points and Classifying Local Maxima and Minima
Don Byrd, rev. 25 Oct. 2011
To find and classify critical points of a function f(x)
1. Take the derivative f ’(x) .
2. Find the critical points by setting f ’ equal to 0, and solving for x .
To finish the job, use either the first derivative test or the second derivative test.
Via First Derivative Test
3. Determine if f ’ is positive (so f is increasing) or negative (so f is decreasing) on both sides
of each critical point.
• Note that since all that matters is the sign, you can check any value on the side you
want; so use values that make it easy!
• Optional, but helpful in more complex situations: draw a sign diagram.
For example, say we have critical points at 11/8 and 22/7. It’s usually easier to evaluate
functions at integer values than non-integers, and it’s especially easy at 0. So, for a value
to the left of 11/8, choose 0; for a value to the right of 11/8 and the left of 22/7, choose 2;
and for a value to the right of 22/7, choose 4. Let’s s ay f ’(0) = –1; f ’(2) = 5/9; and f ’(4) =
5. Then we could draw this sign diagram:
Value of x 11 22
Sign of f ’(x) negative positive positive
4. Apply the first derivative test (textbook, top of p. 172, restated in terms of the derivative):
If f ’ changes from negative to positive: f has a local minimum at the cri