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Cogito

Calculus · Chapter 6 · Lesson 2

Practice Worksheet

The Second Derivative and Concavity

How the slope itself is changing.

12 problems · about 22 minutes · FUN-4.A

NameDate

Figure — use these to answer the problems

A cubic changes concavity exactly once. Find the point where the bend flips direction.
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y = 1x³ − 3x + 0
  • Point(0, 0)
  • Slope of the tangent-3
a1
b-3
Point of tangency0
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. f′(4) = 0 and f″(4) = −3. Max or min? 1 max, 2 min.

    Answer

  2. What does an inflection point require?

    • The second derivative must change sign.
    • The second derivative must equal zero.

  3. f″(3) = 4. Concave up or down? 1 up, 2 down.

    Answer

  4. f′(5) = 0 and f″(5) = 2. Max or min? 1 max, 2 min.

    Answer

  5. f(x) = x², so f″(x) = 2. Concave up or down everywhere? 1 up, 2 down.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. f(x) = x³ − 3x, so f″(x) = 6x. What is f″(−2)?

    Answer

  2. f′(1) = 0 and f″(1) = 0. Is the test conclusive? 1 for yes, 0 for no.

    Answer

  3. How many inflection points does a cubic have?

    Answer

Stretch Yourself

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

4 problems

  1. Sort each sign of the second derivative by what it means.

    Write each item under the heading it belongs to: f″ = 3 · f″ = −7 · The slope is increasing · The slope is decreasing

    Concave up

    Concave down

  2. f(x) = x⁴, so f″(x) = 12x². What is f″(2)?

    Answer

  3. The Test

    f′(2) = 0 and f″(2) = −5. Is x = 2 a maximum or minimum? 1 max, 2 min.

    Answer

  4. The Inflection

    f″(x) = 6x. At which x is there an inflection point?

    Answer