Critical Points of cos y & cos 2x - Why the Difference?

[rsaad]

for cos y the critical points are n*pi - 0.5*pi
for cos 2x, the critical points should be n/2 * pi - 0.25*pi
because if I set y=2x = n*pi - 0.5*pi , I get the eq in red for x.
However if I do it graphically, I just get n*pi - 0.25*pi.
The question: why algebraically I am constrained to use the even number n but graphically, there is no restriction on n?

 

[mfb]

I think your critical points for cos y are wrong. Do you mean sin?

The question: why algebraically I am constrained to use the even number n but graphically, there is no restriction on n?

Answer: At least one of them is wrong.

 

[pwsnafu]

I think you have critical point and root mixed up.