Science:Math Exam Resources/Courses/MATH100/December 2018/Question 01 (d)

MATH100 December 2018

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Question 01 (d)
The function $f$ has a derivative defined and continuous for all $x$, and critical points only for $x\in \{0,1,2\}.$ If we know that $f(-1)=1,f(0)=0,f(1)=1,f(2)=2$ and $f(3)=1$, list all points ($x$ and $y$ coordinates) where $f$ has a local maximum.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
If a function has a derivative defined and continuous everywhere, then its local maxima and minima must occur at points where the derivative is zero.

Hint 2
It might help to sketch a graph of the function, recalling that since the graph has derivative defined and continuous, the derivative can only change sign at the critical points.

Checking a solution serves two purposes: helping you if, after having used all the hints, 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.

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. Since $f$ has a continuous derivative everywhere, the only candidates for the local maxima and minima are its critical points. Note that $f(2)$ is larger than $f(1)$ and $f(3)$, so by the Mean Value Theorem the function has a local minimum at $x=2$. This rules out the critical point $x=1$ since by the MVT the function is increasing on the interval $(1,2)$. Moreover, since $f(0)$ is smaller than $f(0)$ and $f(1)$, a similar application of the MVT shows that it is a local minimum. We conclude that the only local maximum is at $\color {blue}(2,2)$.

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