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 ${\textstyle f}$ is ${\textstyle \{x\neq 0\}}$ and the derivative of ${\textstyle f}$ is

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

Then, we can easily see that the only critical point of ${\textstyle f}$ is ${\textstyle x=2}$ (because ${\textstyle f'(2)=0}$. Note that ${\textstyle x=0}$ 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 ${\textstyle f}$ is increasing on ${\textstyle (0,2)}$, while it is decreasing on ${\textstyle (2,\infty )}$, by the first derivative test in the Hint, the point ${\textstyle \color {blue}(2,1/4)}$ is the local maximum.