Finding critical Numbers with multiple trig functions

[cj122]

I need to find all the critical numbers for the function
f(x)=(sinx)^2+cosx where 0 < x < pi

I was able to get the dirivitive of sinx(2cosx - 1)

What do I do from this step forward to each my critical numbers?

 

[DrPhil]

I need to find all the critical numbers for the function
f(x)=(sinx)^2+cosx where 0 < x < pi

I was able to get the dirivitive of sinx(2cosx - 1)

What do I do from this step forward to each my critical numbers?

In order to find critical values, you have to set the derivative to 0. The expression is 0 if (and only if) one factor or the other is 0.

 

[cj122]

In order to find critical values, you have to set the derivative to 0. The expression is 0 if (and only if) one factor or the other is 0.

with non trig. problems that simple enough, just devided and subtract away the excess coefficents. but how would one do that with trig functions?

 

[DrPhil]

with non trig. problems that simple enough, just devided and subtract away the excess coefficents. but how would one do that with trig functions?

Your derivative has two factors:

\(\displaystyle f'(x) = \sin(x) \times (2\ \cos(x) - 1) = 0\)

The expression is 0 if (and only if) one factor or the other is 0. Set each factor to 0 and solve for x.