 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/20/b/i/solution

The curve has

x \neq = 0

and can be written

y = 1/ x - x

. It is a hyperbola with vertical asymptote

x = 0

and oblique asymptote

y = - x

, crossing the horizontal axis at

(\pm 1, 0)

. It is symmetric under

(x, y) \mapsto (- x, - y)

On each branch

d y / d x = - 1/ x^{2} - 1 < 0

. For

x > 0

, the branch descends from

+ \infty

x ↓ 0

, passes through

(1, 0)

, and approaches

y = - x

from above as

x \to + \infty

. The negative branch follows by the central symmetry.

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