Science:Math Exam Resources/Courses/MATH104/December 2015/Question 06 (a)

MATH104 December 2015

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Question 06 (a)
Below is a graph of $f(x).$ (i) What is the global maximum value? (ii) State the $x$-value(s) where a local minimum occurs.

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
Recall that a global maximum is a value $f(x^{*})$ of $f$ for which $f(x^{*})\geq f(x)$ for all $x$ in the domain of $f.$

Hint 2
Recall that a local minimum is a value $f(x^{*})$ of $f$ for which $f(x^{*})\leq f(x)$ for all $x$ in some open interval centred at $x^{*}$, i.e., an interval of the form $(x^{*}-\varepsilon ,x^{*}+\varepsilon ).$

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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. (i) Visually speaking, we can find a global maximum of $f$ by searching for the highest "y-value" attained by its graph (see hint 1 for a precise definition). We see that it is $3$ (which occurs when $x=4$, although this information is not necessary here). Hence $\max _{x}f(x)={\color {blue}3}.$ (ii) Local minima of $f$ appear as 'valleys' in its graph (see hint 2 for a precise definition). We see that local minima occur at $x={\color {blue}2},{\color {blue}7}.$ Note: The point $x=9$ looks tempting, but note that $f(9)=2$, which is certainly not less than or equal to values of $f$ nearby. We also cannot choose something "just to the left of 9", e.g., $x=8.9$, since we will always be able to find a value of $x$ closer to 9 for which $f(x)$ is smaller (e.g., $x=8.99$).