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Cogito

AP Calculus AB · Chapter 5 · Lesson 1

Practice Worksheet

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

NameDate

Figure — use these to answer the problems

Move the point onto a turn and watch the tangent flatten.
-8-8-6-6-4-4-2-222446688
y = 1x³ − 3x + 0
  • Point(0, 0)
  • Slope of the tangent-3
a1
b-3
c0
x0
Figure 1

Warm Up

Straightforward practice. Get the method working first.

4 problems

  1. f'(x) = 2x + 14. Critical point at which x?

    Answer

  2. f'(x) = 2x − 10. Critical point at which x?

    Answer

  3. f'' = 3 at a critical point. 1 for max, 2 for min.

    Answer

  4. f'(x) = 3x². Critical point at which x?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Is that critical point of x³ an extremum? 1 for yes, 0 for no.

    Answer

  2. Sort each first-derivative behaviour by what it means at the critical point.

    Write each item under the heading it belongs to: Positive then negative · Negative then positive · Positive on both sides

    Maximum

    Minimum

    Neither

  3. f'(x) = x² − 9. How many critical points?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. f'' = 0 at a critical point. Does the second derivative test decide it? 1 for yes, 0 for no.

    Answer

  2. f'(x) = 4x + 12. Critical point at which x?

    Answer

  3. The Flat Tangent

    f'(x) = 2x − 8. At what x is the critical point?

    Answer

  4. The Test

    At a critical point, f'' = −5. Maximum or minimum? 1 for max, 2 for min.

    Answer