  ### Home > CALC > Chapter 6 > Lesson 6.2.1 > Problem6-63

6-63.

Find the first and second derivatives and use them to test for maxima and minima on the given interval. Remember to check the endpoints!

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

This problem is asking you to find EXTREME VALUES on a closed domain.
Extreme values (global maxima and global minima) can exist where $y^\prime= 0$ or where $y^\prime =$ DNE. Find extrema candidates where $y^\prime = 0$. Remember to only consider candidates within the given domain: $[0, π]$.
$y^\prime = 2\operatorname{cos}x − 2\operatorname{sin}x = 2(\operatorname{cos}x − \operatorname{sin}x)$
$0 = 2(\operatorname{cos}x − \operatorname{sin}x)$
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Find more extrema candidates where $y^\prime =$ DNE. In other words, where is y non-differentiable? Are there endpoints, jumps, holes, cusps or vertical tangents? If so, these are extrema candidates. There are endpoint candidates at $x = 0$ and $x = π$. $3$ extrema candidates have been identified.

Test each candidate to see who has the most extreme (highest and lowest) $y$-values.
The 'winners' are the global max and the global min.

$y(0) =$ _________

$y\left (\frac{\pi}{4} \right )=$ _________

$y(π) =$ _________