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Cogito

Multivariable Calculus · Chapter 6 · Lesson 1

Practice Worksheet

Critical Points and the Second Derivative Test

Flat spots, and how to tell which kind you found.

12 problems · about 24 minutes · F-IF.B.4, F-IF.C.7

NameDate

Figures — use these to answer the problems

The origin here is flat, yet it is neither a peak nor a pit. That is what D < 0 detects.
  • The surfacez = a(x² − y²)
  • Height at the origin0
a1
b1
Figure 1
The same saddle from above. Contours pulling apart in opposite directions is the signature.
a1
b1
Figure 2

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. fₓₓ = 3, fᵧᵧ = 12, fₓᵧ = 6. What is D?

    Answer

  2. What is a saddle point?

    • A flat point that is a minimum one way and a maximum another way.
    • The highest point on the surface.

  3. fₓₓ = 6, fᵧᵧ = 6, fₓᵧ = 0. What is D?

    Answer

  4. D = 36 and fₓₓ = 6. Enter 1 for minimum, 2 for maximum, 3 for saddle.

    Answer

  5. D = 36 and fₓₓ = −6. Enter 1 for minimum, 2 for maximum, 3 for saddle.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. fₓₓ = 1, fᵧᵧ = 1, fₓᵧ = 3. What is D?

    Answer

  2. D = 0. Does the test give an answer? 1 yes, 0 no.

    Answer

  3. At a critical point, what is the magnitude of the gradient?

    Answer

Stretch Yourself

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

4 problems

  1. Match each case to its classification.

    Draw a line from each item on the left to its match on the right.

    • D > 0 and fₓₓ > 0
    • D > 0 and fₓₓ < 0
    • D < 0
    • Local minimum
    • Local maximum
    • Saddle point

  2. f = x² − y². What is D at the origin, given fₓₓ = 2, fᵧᵧ = −2, fₓᵧ = 0?

    Answer

  3. The Test

    fₓₓ = 2, fᵧᵧ = 8 and fₓᵧ = 3. What is D?

    Answer

  4. The Verdict

    D = −5 at a critical point. Enter 1 for minimum, 2 for maximum, 3 for saddle.

    Answer