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

1 Answer
Jun 10, 2018

Decreasing at #x=pi/12#

Explanation:

#f(x)=cos(3x-pi/6)+2sin(4x-(3pi)/4)#
#f'(x)=-3sin(3x-pi/6)+8cos(4x-(3pi)/4)#
#f''(x)=-9cos(3x-(pi)/6)-32sin(4x-(3pi)/4)#

For #x=pi/12#,

Sub the #x# into the second derivative ie #f''(x)# 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''(pi/12)=-9cos(pi/4-pi/6)-32sin(pi/3-(3pi)/4)#

#f''(pi/12)=-9cos(pi/12)-32sin(-(5pi)/12)#

#f''(pi/12)=22.22 >0#

Therefore, the graph is decreasing at #x=pi/12#

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

From the graph, you can see that at #x=pi/12# which is approximately 0.26, the graph at that point is indeed decreasing.