 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/14/b/iii/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 14 / b / iii / Solution by

Codex  0 2026-10-18

For

a = - 1

, the local minimum at the origin has

f = 1

. Its full level set is

y^{2} = \frac{x ^{4}}{2} - x^{2} .

The connected component containing the minimum is just the isolated origin. The same level also has the outer branches

y = \pm x^{4} /2 - x^{2}

for

∣ x ∣ \geq 2

; these are different connected components, not curves approaching the origin.

Both saddle points

(\pm 1, 0)

have

f = e^{1/4}

. Their common level set factors as

y^{2} = \frac{x ^{4}}{2} - x^{2} + \frac{1}{2} = \frac{( x ^{2} - 1 ) ^{2}}{2} .

Its explicit equations are therefore

y = \frac{x ^{2} - 1}{2} or y = - \frac{x ^{2} - 1}{2} .

The two parabolas cross at each saddle, explaining the change of contour connectivity there. Each displayed equation can be restricted to the square of part (ii) if only that sketch window is wanted.

 Read the full article

 New to topics? Read the docs here!