Cogito
Multivariable Calculus · Chapter 6 · Lesson 1
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
What this lesson teaches
The student finds critical points and classifies them with the discriminant test.
- A critical point has both partial derivatives zero, so the tangent plane is horizontal.
- D = fₓₓfᵧᵧ − (fₓᵧ)² sorts peaks and pits from saddles.
- D < 0 means a saddle; D = 0 means the test cannot decide.
Warm Up
Straightforward practice. Get the method working first.
5 problemsfₓₓ = 3, fᵧᵧ = 12, fₓᵧ = 6. What is D?
Answer 0
Why 0, so the test is inconclusive.
What is a saddle point?
Answer A flat point that is a minimum one way and a maximum another way.
Why Up in one direction, down in another.
fₓₓ = 6, fᵧᵧ = 6, fₓᵧ = 0. What is D?
Answer 36
Why 36 − 0.
D = 36 and fₓₓ = 6. Enter 1 for minimum, 2 for maximum, 3 for saddle.
Answer 1
Why Positive D, positive curvature.
D = 36 and fₓₓ = −6. Enter 1 for minimum, 2 for maximum, 3 for saddle.
Answer 2
Why Curving downward.
Build It Up
The same ideas with more to keep track of.
3 problemsfₓₓ = 1, fᵧᵧ = 1, fₓᵧ = 3. What is D?
Answer -8
Why 1 − 9.
D = 0. Does the test give an answer? 1 yes, 0 no.
Answer 0
Why It declines.
At a critical point, what is the magnitude of the gradient?
Answer 0
Why Both partials vanish.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each case to its classification.
Answer D > 0 and fₓₓ > 0 → Local minimum; D > 0 and fₓₓ < 0 → Local maximum; D < 0 → Saddle point
Why The sign of D decides extreme against saddle first.
f = x² − y². What is D at the origin, given fₓₓ = 2, fᵧᵧ = −2, fₓᵧ = 0?
Answer -4
Why 2 × (−2).
The Test: fₓₓ = 2, fᵧᵧ = 8 and fₓᵧ = 3. What is D?
Answer 7
Why 7.
The Verdict: D = −5 at a critical point. Enter 1 for minimum, 2 for maximum, 3 for saddle.
Answer 3
Why A saddle.