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

In summary, there is a difference between the critical points and roots for cos y and cos 2x. Algebraically, the critical points for cos y are n*pi - 0.5*pi and for cos 2x, they should be n/2 * pi - 0.25*pi. However, when graphing, there is no restriction on the value of n for the critical points. This raises the question of why there is a constraint on n algebraically but not graphically.
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?

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.

I think you have critical point and root mixed up.

## 1. What are critical points in mathematics?

Critical points are points on a function where the derivative is equal to zero or is undefined. They are important in determining the behavior and shape of a function.

## 2. How do you find critical points of cos y?

To find the critical points of cos y, you need to take the derivative of cos y with respect to y. This will give you -sin y. Set -sin y equal to zero and solve for y. The resulting values of y are the critical points of cos y.

## 3. How do you find critical points of cos 2x?

To find the critical points of cos 2x, you need to take the derivative of cos 2x with respect to x. This will give you -2sin 2x. Set -2sin 2x equal to zero and solve for x. The resulting values of x are the critical points of cos 2x.

## 4. Why is there a difference in the critical points of cos y and cos 2x?

The difference in the critical points of cos y and cos 2x is due to the fact that the functions are different. While cos y is a function of y, cos 2x is a function of x. This means that the critical points will be different since they are determined by taking the derivative with respect to a different variable.

## 5. How do critical points affect the graph of a function?

Critical points can affect the graph of a function in a few ways. They can indicate the presence of local maxima or minima, points of inflection, or discontinuities. They can also help determine the concavity of a function and the intervals where the function is increasing or decreasing.

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