Question

What is the maximum value that the graph of `y=cos x` assumes?

Answer 1

`y=|A|cos(x)`, where `|A|` is the amplitude.

The cosine function oscillates between the values -1 to 1.

The amplitude of this particular function is understood to be 1.

`|A| = 1`

`y=1*cos(x)=cos(x)`

Answer 2

The maximum value of the function `cos(x)` is `1`.

This result can be easily obtained using differential calculus.

First, recall that for a function `f(x)` to have a local maximum at a point `x_0` of it's domain it is necessary (but not sufficient) that `f^prime(x_0)=0`. Additionally, if `f^((2)) (x_0)<0` (the second derivative of f at the point `x_0` is negative) we have a local maximum.

For the function `cos(x)`:

`d/dx cos(x)=-sin(x)`

`d^2/dx^2 cos(x)=-cos(x)`

The function `-sin(x)` has roots at points of the form `x=n pi`, where `n` is an integer (positive or negative).

The function `-cos(x)` is negative for points of the form `x= (2n+1) pi` (odd multiples of `pi`) and positive for points of the form `2n pi` (even multiples of `pi`).

Therefore, the function `cos(x)` has all it's maximums at the points of the form `x=(2n+1)pi`, where it takes the value `1`.