# Determine the intervals on which f is increasing or decreasing and whether each of the critical points is a local maximum, minimum, or neither.   F(x) = 1/2sin^2 + cosx on [0,2pi] Determine the intervals where the function is increasing or decreasing and the type of critical points found on the interval `[0,2pi]` for the function: `f(x)=1/2 sin^2x+cosx`

The critical points will be found where the first derivative is zero.

`f'(x)=sinxcosx-sinx`

`f'(x)=0 ==> sinx(cosx-1)=0` Then:

`sinx=0 ==> x=0,pi,2pi`

`cosx=1==>x=0,2pi`

For `0<x<pi,f'(x)<0` so...

Determine the intervals where the function is increasing or decreasing and the type of critical points found on the interval `[0,2pi]` for the function: `f(x)=1/2 sin^2x+cosx`

The critical points will be found where the first derivative is zero.

`f'(x)=sinxcosx-sinx`

`f'(x)=0 ==> sinx(cosx-1)=0` Then:

`sinx=0 ==> x=0,pi,2pi`

`cosx=1==>x=0,2pi`

For `0<x<pi,f'(x)<0` so the function is decreasing on this interval.

For `pi<x<2pi,f'(x)>0` so the function is increasing on this interval.

The function has a maximum at `x=0,x=2pi` and a minimum at `x=pi`

The graph:

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