Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 27(b)

MATH100 A December 2023

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[hide] Question 27(b)
Determine the intervals where $f$ is increasing and the intervals where $f$ is 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?

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
Recall that a function $f$ is increasing (or decreasing) at every $x$ for which $f'(x)$ is positive (or negative).

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

[show] Solution

We analyze the sign of $f'(x)=-4e^{-x/2}(x-2)$. Note that $f'(x)=ABC$, where $A=-4$, $B=e^{-x/2}$ and $C=x-2$. A product of 3 numbers is positive if the numbers are either all positive or if two of them are negative. Since $A$ is negative for all $x$, it follows that $f'(x)$ is positive if and only if precisely one of $B$ or $C$ is negative. $B$ is positive for all $x$, so we only need to analyze the sign of $C$, which is clearly positive for $x<2$ and negative for $x>2$. Putting it all together, we find that $f'(x)$ is positive if $x\in [0,2)$ and negative if $x\in (2,\infty )$, so $f$ is increasing on the interval $[0,2)$ and decreasing on the interval $(2,\infty )$.