Critical Points and Optimisation in Two Variables
Find points where the gradient vanishes and classify them with the second derivative test, including the saddle point, which has no one-variable analogue.
What a learner can do afterwards
- Locate all critical points by solving the gradient equal to zero
- Classify a critical point with the discriminant of the second partials
- Describe a saddle and say why one-variable tests miss it
1 · Read
Critical points are where the gradient vanishes, the flat candidates for peaks, pits, and saddles. Solve fx = 0 and fy = 0 together. For x squared minus 2x + y squared + 6y + 10: 2x minus 2 = 0 gives x = 1, 2y + 6 = 0 gives y = minus 3, and the value there is 0.
The discriminant D = fxx fyy minus fxy squared sorts each candidate. For x squared + y squared: fxx = 2, fyy = 2, fxy = 0, so D = 4 with fxx > 0, a local minimum at the origin. For x squared minus y squared: D = minus 4, a saddle.
Read the table: D > 0 with fxx > 0 is a min, D > 0 with fxx < 0 is a max, D < 0 is a saddle. Jay is right that negative D always means a saddle. Keep this table beside you until the cases feel routine.
The saddle is the new animal here: up in one slice and down in another. Single variable tests only check axis slices, so they can miss the mix entirely. When D goes negative, picture a mountain pass, not a peak or a pit.
Solve gradient zero for candidates, then let D and the sign of fxx deliver the verdict.
More to read
4.7 Maxima/Minima Problems
Critical points are where the gradient vanishes, the flat candidates for peaks, pits, and saddles. Solving fx = 0 and fy = 0 together gives the full list, and each one needs classification. The discriminant D = fxx fyy minus fxy squared sorts them: positive with fxx positive is a min, positive with fxx negative is a max, negative is a saddle. The saddle is the new animal here, up in one slice and down in another, which single variable tests cannot see. Work the examples with the D table beside you until the cases feel routine.
Reasoning behind second partial derivative test
This article explains why the discriminant works instead of just stating the cases. The idea is to approximate the surface by its quadratic part near the critical point and read the bowl shape off that. Positive definite bowls hold minima, negative definite bowls hold maxima, and mixed bowls are saddles. Read it once for intuition even if you compute with the textbook table. Knowing the why makes the sign cases hard to mix up.
Examples: Second partial derivative test
A set of worked classifications from gradient zero through D to the verdict. Follow each one with pen in hand, hiding the next line before it is revealed. Notice how the algebra of solving the gradient system is half the battle. The classification step at the end is then quick. Use these as models for the practice items on locating and sorting critical points.
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2 · Watch
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Optional connections
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16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.