 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/8/solution

Differentiate the two powers to locate the stationary point:

y^{'} = \frac{x - 2}{x ^{3}} .

For

x > 0

, this is negative before

2

and positive afterwards, so the global minimum is

x = 2, y = - \frac{1}{4} .

The curve crosses the horizontal axis at

x = 1

. Its limits are

+ \infty

x \to 0^{+}

and

0

from below as

x \to \infty

, giving the vertical asymptote

x = 0

and horizontal asymptote

y = 0

. These features determine the required sketch.

 New to topics? Read the docs here!