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Section 4-4 : Finding Absolute Extrema

11. Determine the absolute extrema of \(f\left( y \right) = \sin \left( {{ \displaystyle \frac{y}{3}}} \right) + { \displaystyle \frac{2y}{9}}\) on \(\left[ { - 10,15} \right]\).

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Hint : Just recall the process for finding absolute extrema outlined in the notes for this section and you’ll be able to do this problem!
Start Solution

First, notice that we are working with the sine function and this is continuous everywhere and so will be continuous on the given interval. Recall that this is important because we now know that absolute extrema will in fact exist by the Extreme Value Theorem!

Now that we know that absolute extrema will in fact exist on the given interval we’ll need to find the critical points of the function.

Given that the purpose of this section is to find absolute extrema we’ll not be putting much work/explanation into the critical point steps. If you need practice finding critical points please go back and work some problems from that section.

Here are the critical points for this function.

\[f'\left( y \right) = {\displaystyle \frac{1}{3}}\cos \left( {{\displaystyle \frac{y}{3}}} \right) + {\displaystyle \frac{2}{9}} = 0\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\cos \left( {{\displaystyle \frac{y}{3}}} \right) = - {\displaystyle \frac{2}{3}}\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\,{\displaystyle \frac{y}{3}} = {\cos ^{ - 1}}\left( { - {\displaystyle \frac{2}{3}}} \right) = 2.3005\] \[\begin{array}{*{20}{c}}{{\displaystyle \frac{y}{3}} = 2.3005 + 2\pi n}\\{{\displaystyle \frac{y}{3}} = 3.9827 + 2\pi n}\end{array}\hspace{0.25in} \Rightarrow \hspace{0.25in}\,\,\,\,\begin{array}{*{20}{c}}{y = 6.9016 + 6\pi n}\\{y = 11.9481 + 6\pi n}\end{array}\,\,\,\,\,\,\,\,n = 0, \pm 1, \pm 2, \pm , \ldots \]

If you need some review on solving trig equations please go back to the Review chapter and work some of the problems the Solving Trig Equations sections.

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Now, recall that we actually are only interested in the critical points that are in the given interval and so, in this case, the critical points that we need are,

\[y = - 6.9016,\,\,\,\,\,\,y = 6.9016,\,\,\,\,\,\,\,\,y = 11.9481\]

Note that we got these values by plugging in values of \(n\) into the solutions above and checking the results against the given interval.

Show Step 3

The next step is to evaluate the function at the critical points from the second step and at the end points of the given interval. Here are those function evaluations.

\[\begin{array}{c}f\left( { - 10} \right) = - 2.0317\,\,\,\,\,\,f\left( { - 6.9016} \right) = - 2.2790\,\,\,\,\,f\left( {6.9016} \right) = 2.2790\\ f\left( {11.9481} \right) = 1.9098\,\,\,\,\,f\left( {15} \right) = 2.3744\end{array}\] Show Step 4

The final step is to identify the absolute extrema. So, the answers for this problem are then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*}{\mbox{Absolute Maximum : }}& 2 {\mbox{.3744 at }}y = 15\\ {\mbox{Absolute Minimum : }} & - 2.2790{\mbox{ at }}y = - 6.9016\end{align*}}\]

Note the importance of paying attention to the interval with this problem. Without an interval we would have had (literally) an infinite number of critical points to check. Also, without an interval (as a quick graph of the function would show) there would be no absolute extrema for this function.