Cogito
AP Calculus AB · Chapter 5 · Lesson 1
Critical Points and Extrema
Where a function turns, and how to tell.
11 problems · about 25 minutes · AP Calculus AB 5.3, 5.4, 5.5
What this lesson teaches
The student finds critical points and classifies them with the first and second derivative tests.
- Critical points are where f' is zero or undefined.
- Every extremum sits at a critical point, but not every critical point is an extremum.
- The first test watches for a sign change; the second reads concavity and is silent at zero.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf'(x) = 2x + 14. Critical point at which x?
Answer -7
Why x = −7.
f'(x) = 2x − 10. Critical point at which x?
Answer 5
Why Set to zero.
f'' = 3 at a critical point. 1 for max, 2 for min.
Answer 2
Why Concave up.
f'(x) = 3x². Critical point at which x?
Answer 0
Why 3x² = 0.
Build It Up
The same ideas with more to keep track of.
3 problemsIs that critical point of x³ an extremum? 1 for yes, 0 for no.
Answer 0
Why The sign never changes.
Sort each first-derivative behaviour by what it means at the critical point.
Answer Maximum: Positive then negative · Minimum: Negative then positive · Neither: Positive on both sides
Why A sign change is what makes a turn.
f'(x) = x² − 9. How many critical points?
Answer 2
Why x = 3 and x = −3.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsf'' = 0 at a critical point. Does the second derivative test decide it? 1 for yes, 0 for no.
Answer 0
Why The test is silent there.
f'(x) = 4x + 12. Critical point at which x?
Answer -3
Why 4x = −12.
The Flat Tangent: f'(x) = 2x − 8. At what x is the critical point?
Answer 4
Why x = 4.
The Test: At a critical point, f'' = −5. Maximum or minimum? 1 for max, 2 for min.
Answer 1
Why Maximum.