# Is #f(x)= cos(x+(5pi)/4) # increasing or decreasing at #x=-pi/4 #?

See explanation.

In the example provided, we have:

graph{<sin(x+(5pi)/4)+y^2)-0.01)=0 [-4, 4, -2, 2]}

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To determine whether ( f(x) = \cos\left(x + \frac{5\pi}{4}\right) ) is increasing or decreasing at ( x = -\frac{\pi}{4} ), we need to examine the derivative of the function at that point.

The derivative of ( \cos(x) ) is ( -\sin(x) ). Thus, the derivative of ( f(x) ) with respect to ( x ) is ( -\sin\left(x + \frac{5\pi}{4}\right) ).

At ( x = -\frac{\pi}{4} ), we need to evaluate ( -\sin\left(-\frac{\pi}{4} + \frac{5\pi}{4}\right) ).

( -\frac{\pi}{4} + \frac{5\pi}{4} = \pi )

( \sin(\pi) = 0 )

Since the sine function equals ( 0 ) at ( x = \pi ), the derivative of ( f(x) ) is ( 0 ) at ( x = -\frac{\pi}{4} ).

Therefore, the function ( f(x) ) is neither increasing nor decreasing at ( x = -\frac{\pi}{4} ).

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