CPM Homework Help : CALC3RD Problem 6-63

Write the first and second derivatives for the function below and use them to test for extrema over the given interval. Remember to check the endpoints!

$y = 2 \sin(x) + 3 \cos(x) \text{ over } [0, π]$

Hint:

This problem is asking you to find EXTREME VALUES on a closed domain.
Extreme values (global maxima and global minima) can exist where $y' = 0$ or where $y' =$ DNE.

Step 1:

Find the first and second derivatives. Locate extrema candidates where $y' = 0$ . Remember to only consider candidates within the given domain: $[0, π]$ .
$y'=2\cos(x)−3\sin(x)$
$y''=-2\sin(x)−3\cos(x)$
$y'=0$ when $2\cos(x)=3\sin(x)$

Therefore,
$\frac{2}{3}=\tan\left(x\right);\ x=\arctan\left(\frac{2}{3}\right)\$ ; about $0.588$

Step 2:

Find y(0.588). This would be a local maximum. Find the y-value of each of the endpoints. Now determine the extrema.