Science:Math Exam Resources/Courses/MATH110/April 2016/Question 05 (f)/Solution 1

From the part (a) and (d) of this question, we know that the domain of $f$ is $\{x\ne 0\}$ and the derivative of $f$ is

$$f^{\prime }(x)=\frac{2-x}{x^3}.$$

Then, we can easily see that the only critical point of $f$ is $x=2$ (because $f^{\prime }(2)=0$. Note that $x=0$ is excluded because it is not in the domain.)

Since the local extreme(s) only occur at the critical point(s), $x=2$ is the only candidate for the local extreme.

In the part (e), we observe that $f$ is increasing on $(0,2)$, while it is decreasing on $(2,\mathrm{\infty })$, by the first derivative test in the Hint, the point $(2,1/4)$ is the local maximum.