---
title: "Critical Points and Optimisation in Two Variables"
description: "Find points where the gradient vanishes and classify them with the second derivative test, including the saddle point, which has no one-variable analogue."
canonical: https://lightmysky.com/learn/mathematics/critical-points-and-optimisation-in-two-variables-mt_4qB5pAGeuR
source: https://lightmysky.com/learn/mathematics/critical-points-and-optimisation-in-two-variables-mt_4qB5pAGeuR.md
retrieved: 2026-09-12
---

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# 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.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/critical-points-and-optimisation-in-two-variables-mt_4qB5pAGeuR

## Ready when they can

- 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

## Lesson: Flat points: peaks, pits, and saddles

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.

**Example.** 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.

**Tip.** 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.

**Recap.** Solve gradient zero for candidates, then let D and the sign of fxx deliver the verdict.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Optimisation with Calculus](https://lightmysky.com/learn/mathematics/optimisation-with-calculus-mt_9I0ql39Lzw)
- [Directional Derivatives and the Gradient](https://lightmysky.com/learn/mathematics/directional-derivatives-and-the-gradient-mt_LW_KckY5Ad)

## Opens up

- [Lagrange Multipliers](https://lightmysky.com/learn/mathematics/lagrange-multipliers-mt_0RPOBWZ1gs)