Question

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

Answer 1

It's decreasing.

Explanation:

The answer lies in the first derivative: if it is positive at `x=-pi/6`, the function will be increasing, otherwise it will be decreasing.

So, the first derivative is

`1-12cos(3x-pi/4)*3 = 1-36cos(3x-pi/4)`

Evaluate at `x=-pi/6`:

`1-36cos(3(pi/6)-pi/4) = 1-36cos(pi/2-pi/4)`
`=1-36cos(pi/4) = 1-36sqrt(2)/2`
`=1-18sqrt(2)`

which is clearly negative. So, the function is decreasing, as you can see in the graph.