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

Decreasing at

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

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To determine if ( f(x) = \cos(3x - \frac{\pi}{6}) + 2\sin(4x - \frac{3\pi}{4}) ) is increasing or decreasing at ( x = \frac{\pi}{12} ), we need to find the first derivative ( f'(x) ) and evaluate it at ( x = \frac{\pi}{12} ).

- Find ( f'(x) ):

Using the chain rule and the derivatives of ( \cos ) and ( \sin ):

[ f'(x) = -3\sin(3x - \frac{\pi}{6}) + 8\cos(4x - \frac{3\pi}{4}) ]

- Evaluate ( f'(x) ) at ( x = \frac{\pi}{12} ):

[ f'\left(\frac{\pi}{12}\right) = -3\sin\left(\frac{\pi}{4}\right) + 8\cos\left(\frac{\pi}{3}\right) ]

Using trigonometric values: [ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} ] [ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} ]

Substituting these values:

[ f'\left(\frac{\pi}{12}\right) = -3 \times \frac{\sqrt{2}}{2} + 8 \times \frac{1}{2} ] [ f'\left(\frac{\pi}{12}\right) = -\frac{3\sqrt{2}}{2} + 4 ]

Now, to determine if ( f(x) ) is increasing or decreasing at ( x = \frac{\pi}{12} ):

If ( f'\left(\frac{\pi}{12}\right) > 0 ), then ( f(x) ) is increasing. If ( f'\left(\frac{\pi}{12}\right) < 0 ), then ( f(x) ) is decreasing.

Calculating ( f'\left(\frac{\pi}{12}\right) ):

[ f'\left(\frac{\pi}{12}\right) = -\frac{3\sqrt{2}}{2} + 4 ]

Since ( -\frac{3\sqrt{2}}{2} ) is negative and ( 4 ) is positive:

[ f'\left(\frac{\pi}{12}\right) < 0 ]

Thus, ( f(x) ) is decreasing at ( x = \frac{\pi}{12} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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