Science:Math Exam Resources/Courses/MATH110/April 2018/Question 07 (d)

MATH110 April 2018

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[hide] Question 07 (d)
Let $f(x)={\frac {e^{x}}{x}}$. (d) Find the $x$-coordinates of all local maxima and minima of $f$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

[show] Hint
If a local extremum of a function $f$ occurs at $x=a$, either $f'(a)=0$ or $f'(a)$ does not exist.

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[show] Solution

Recall that the point $x=0$ is not in the domain of $f$. (See the solution of part (c).) Since $f'(x)={\frac {(x-1)e^{x}}{x^{2}}}$ (in the solution of part (c)) is defined for any $x\neq 0$ and it vanishes at $x=1$, the only candidate of the points where local extrema occur is $x=1$ according to the Hint. From part (c), $f$ is increasing on $(1,\infty )$ and decreasing on $(0,1)$, $f$ has its local minimum at $x=1$. On the other hand, we don't have any local maximum of $f$. In conclusion, the only local extremum of $f$ occurs at $x=1$, which is a local minimum. Answer: There's no local maximum of $f$, while the only local minimum occurs at $\color {blue}x=1$.