Question

Is `f(x)=cos2x-sin^2x` increasing or decreasing at `x=pi/6`?

Answer 1

`f(x)` is decreasing at `pi/6`

Explanation:

To check if this function is increasing or decreasing we should compute `color(blue)(f'(pi/6))`
If
`color(red)(f'(pi/6)<0` then this function is decreasing
`color(red)(f'(pi/6)>0` then this function is increasing

`f(x)=cos2x-sin^2x`
`f'(x)=-2sin2x-2sinxcosx`
`f'(x)=-2sin2x-sin2x`
`f'(x)=-3sin2x`

`color(blue)(f'(pi/6))=-3sin(2*(pi/6))=-3sin(pi/3)=-3*sqrt3/2`
`color(red)(f'(pi/6)=-3sqrt3/2<0`

then this function is decreasing