WORKED EXAMPLE 3· Engineering Mathematics
Extreme Values of a Quartic Two‑Variable Function
A full walkthrough of stationary points and the second‑derivative test, applied to \(f(x,y) = x^4 + y^4 - 2x^2 + 4xy - 2y^2\) — including the case where the test itself stays silent.
- #PartialDerivatives
- #StationaryPoints
- #SecondDerivativeTest
- #CalculusOfTwoVariables
- #InconclusiveCase
Watch first — Example 3
Locating local maxima, minima, and saddle points for a function of two variables — a companion explainer on YouTube.
Before the algebra, the idea is simple: a smooth surface \(z = f(x,y)\) flattens out at its peaks, valleys, and saddles. Set both partial derivatives to zero to find those flat points, then use a second‑derivative test to tell which kind each one is.
📐 The worked example 3
A classic board question, solved step by step.
Examine for extreme values of \(f(x,y) = x^4 + y^4 - 2x^2 + 4xy - 2y^2\)A/M‑23‑R‑21
Step 1 — First partial derivatives
Step 2 — Locate the stationary points
Subtracting the two equations gives \(y = -x\). Substituting into \(x^3 - x + y = 0\):
With \(y = -x\), this gives three stationary points: \((0,\,0)\), \((\sqrt{2},\,-\sqrt{2})\), and \((-\sqrt{2},\,\sqrt{2})\).
Surface of \(f(x,y) = x^4+y^4-2x^2+4xy-2y^2\) — two symmetric wells at \((\pm\sqrt{2},\,\mp\sqrt{2})\) with a flat, inconclusive saddle region at the origin.
Step 3 — Second partial derivatives
Step 4 — Classify each point
| \((0,\,0)\) | \((\sqrt{2},\,-\sqrt{2})\) | \((-\sqrt{2},\,\sqrt{2})\) | |
|---|---|---|---|
| A = \(12x^2-4\) | −4 < 0 | 20 > 0 | 20 > 0 |
| B = 4 | 4 | 4 | 4 |
| C = \(12y^2-4\) | −4 | 20 | 20 |
| \(AC - B^2\) | 0 | 384 > 0 | 384 > 0 |
| Inconclusive | Minimum | Minimum |
At \((0,0)\), \(AC-B^2=0\) — the test gives no verdict there. At the other two points, \(AC-B^2 > 0\) with \(A>0\), so both are minima.
Step 5 — Evaluate the minimum values
💡 Why this matters
Three things worth keeping from this example.
- A stationary point only tells you where the surface is flat — the second‑derivative test tells you what kind of flat it is.
- \(AC - B^2 > 0\) rules out a saddle; the sign of \(A\) (or \(C\)) then decides maximum versus minimum.
- By symmetry, both \((\sqrt2,-\sqrt2)\) and \((-\sqrt2,\sqrt2)\) reach the same minimum value — the surface has two identical wells.
- When \(AC - B^2 = 0\), the second‑derivative test is silent — as at \((0,0)\) here — and a closer look (or a higher‑order check) is needed to classify the point.
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