Question

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

Answer 1

The function is decreasing.

Explanation:

If `f'(-pi/6)<0`, then the function is decreasing at `x=-pi/6`.
If `f'(-pi/6)>0`, then the function is increasing at `x=-pi/6`.

Find `f'(x)`.

`f'(x)=1-12cosx`

Find `f'(-pi/6)`.

`f'(-pi/6)=1-12cos(-pi/6)=1-12(sqrt3/2)=1-6sqrt3`

Since `f'(-pi/6)~~-9.39<0`, the function is decreasing at `x=-pi/6`.