Question

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

Answer 1

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.