# Is f(x)= cos(3x-pi/6)+2sin(4x-(3pi)/4)  increasing or decreasing at x=pi/12 ?

Jun 10, 2018

Decreasing at $x = \frac{\pi}{12}$

#### Explanation:

$f \left(x\right) = \cos \left(3 x - \frac{\pi}{6}\right) + 2 \sin \left(4 x - \frac{3 \pi}{4}\right)$
$f ' \left(x\right) = - 3 \sin \left(3 x - \frac{\pi}{6}\right) + 8 \cos \left(4 x - \frac{3 \pi}{4}\right)$
$f ' ' \left(x\right) = - 9 \cos \left(3 x - \frac{\pi}{6}\right) - 32 \sin \left(4 x - \frac{3 \pi}{4}\right)$

For $x = \frac{\pi}{12}$,

Sub the $x$ into the second derivative ie $f ' ' \left(x\right)$ and if the number is positive ie $> 0$, then the graph is decreasing. If the number is negative ie $< 0$, then the graph is increasing.

$f ' ' \left(\frac{\pi}{12}\right) = - 9 \cos \left(\frac{\pi}{4} - \frac{\pi}{6}\right) - 32 \sin \left(\frac{\pi}{3} - \frac{3 \pi}{4}\right)$

$f ' ' \left(\frac{\pi}{12}\right) = - 9 \cos \left(\frac{\pi}{12}\right) - 32 \sin \left(- \frac{5 \pi}{12}\right)$

$f ' ' \left(\frac{\pi}{12}\right) = 22.22 > 0$

Therefore, the graph is decreasing at $x = \frac{\pi}{12}$

graph{cos(3x-pi/6)+2sin(4x-(3pi)/4) [-10, 10, -5, 5]}

From the graph, you can see that at $x = \frac{\pi}{12}$ which is approximately 0.26, the graph at that point is indeed decreasing.