Question

Let f(x) = cos(x) + 2x.
Where does f have critical points?

Answer 1

f has no critical points.

Explanation:

We are given:

`f(x)=\cos(x)+2x`

A function has critical points whenever its derivative equals 0 or is undefined.

Differentiate the function:

`f'(x)=-\sin(x)+2`

Since this will never be undefined, solve for its zeros:

`0=-\sin(x)+2`

Hence:

`\displaystyle \sin(x)=2`

Recall that the value of sine is always between -1 and 1.

Thus, no real solutions exist.

Therefore, f has no critical points.