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

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Answer 1 · 2018-04-18T18:47:47

#f(x)# is increasing at #x=pi/4.#

We need to find #f'(x)# and evaluate it at #x=pi/4.#

If #f'(pi/4)>0,# #f(x)# is increasing at that point.

If #f'(pi/4)<0, f(x)# is decreasing at that point.

Using the Chain Rule,

#f'(x)=-sin(2x+(3pi)/4)*d/dx(2x+(3pi)/4))#

#f'(x)=-2sin(2x+(3pi)/4)#

#f'(pi/4)=-2sin(pi/2+(3pi)/4)#

#f'(pi/4)=-2sin((5pi)/4)=cancel(-2)(cancel-sqrt2/cancel(2))=sqrt2>0#

#f(x)# is increasing at #x=pi/4.#