A critical point is where both partial derivatives are zero, so the tangent plane is horizontal. Every local extreme is one.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Flat is not enough
A flat spot can be a peak, a pit or a saddle. In one variable there was no third option; in two variables there is.
The discriminant
Compute D = fₓₓfᵧᵧ − (fₓᵧ)². The sign of D and of fₓₓ together decide the classification.
The rules
D > 0 with fₓₓ > 0 is a minimum. D > 0 with fₓₓ < 0 is a maximum. D < 0 is a saddle.
When D is zero
D = 0 tells you nothing. The test declines to answer and other reasoning is needed.
What a saddle is
A saddle is a minimum along one direction and a maximum along another. A mountain pass is the everyday example.
Flat spots
A critical point is where the gradient is zero or undefined. Extrema can only occur there or on the boundary, which reduces an infinite search to a finite list plus a boundary analysis.
Saddle points have no one-variable analogue
A surface can rise in one direction and fall in another at the same flat point. That is a saddle, and it is a genuinely new possibility that two variables introduce.
The second derivative test
The discriminant built from the second partials classifies the point: positive with a positive f_xx gives a minimum, positive with negative gives a maximum, negative gives a saddle, zero is inconclusive.
The boundary must be checked
On a closed bounded region the absolute extrema may lie on the edge, where the gradient need not vanish. Checking the boundary separately is a required step, not an optional refinement.
Step 2: Try It Yourself
Tap and try it out.
- The surfacez = a(x² − y²)
- Height at the origin0
A saddle curves upward along one axis and downward along the other. The origin is a critical point that is neither a maximum nor a minimum.
Contours that separate into opposing pairs mean a saddle: uphill one way, downhill the other.
Step 3: Watch an Example
One step at a time.
Watch Hana Classify a Point
Hana has fₓₓ = 4, fᵧᵧ = 9 and fₓᵧ = 0 at a critical point.
- Step 1
She computes D = 4 × 9 − 0², which is 36.
Step 4: Your Turn
Practice makes it stick.
The Test
Problem 1 of 2
fₓₓ = 2, fᵧᵧ = 8 and fₓᵧ = 3. What is D?
The Verdict
Problem 2 of 2
D = −5 at a critical point. Enter 1 for minimum, 2 for maximum, 3 for saddle.
Classify the Flat Spots
1 of 8
fₓₓ = 6, fᵧᵧ = 6, fₓᵧ = 0. What is D?
2 of 8
D = 36 and fₓₓ = 6. Enter 1 for minimum, 2 for maximum, 3 for saddle.
3 of 8
D = 36 and fₓₓ = −6. Enter 1 for minimum, 2 for maximum, 3 for saddle.
4 of 8
fₓₓ = 1, fᵧᵧ = 1, fₓᵧ = 3. What is D?
5 of 8
D = 0. Does the test give an answer? 1 yes, 0 no.
6 of 8
At a critical point, what is the magnitude of the gradient?
7 of 8
Match each case to its classification.
Tap a card on the left to start.
8 of 8
f = x² − y². What is D at the origin, given fₓₓ = 2, fᵧᵧ = −2, fₓᵧ = 0?
Step 5: Quick Check
Show what you know.
Question 1 of 2
fₓₓ = 3, fᵧᵧ = 12, fₓᵧ = 6. What is D?
Question 2 of 2
What is a saddle point?
What You Learned
- 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.