Science:Math Exam Resources/Courses/MATH104/December 2014/Question 05 (b)

MATH104 December 2014

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Question 05 (b)
Let $f(x)={\frac {e^{x}}{x^{2}}}$. Find the intervals on which $f$ is increasing or decreasing.

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?

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Hint
What are the intervals on which the derivative of $f$ exists and is non-zero?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Remark first that $f$ does not exist at $x=0$; hence, we have three intervals on which $f'(x)$ exists and is non-zero, namely $(-\infty ,0),(0,2),$ and $(2,\infty )$. We create a table with these intervals and the function that make up $f'$ to determine when $f'$ is positive or negative, hence when $f$ is increasing or decreasing. $(-\infty ,0)$ $(0,2)$ $(2,\infty )$ $e^{x}$ + + + $(x-2)$ - - + $x^{3}$ - + + $f'$ + - + $f$ increasing decreasing increasing So $f$ is increasing on the intervals $(-\infty ,0)$ and $(2,\infty )$ and decreasing on $(0,2)$.

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