Question

How do you find the critical numbers for `f(x) = x + 2sinx` to determine the maximum and minimum?

Answer 1

The critical points of a function `f(x)` are the `x` that make `f'(x)=0`

Explanation:

We calculate the derivative `f'(x)=1+2cos(x)`, and now we need to find where `f'(x)=1+2cos(x)=0`. But that means:

`-1=2cos(x)`, and then `cos(x) = -1/2`. Between `0` and `2pi` these points are:

`x=4/6pi=2/3pi` and `8/6pi=4/3pi`,

and all the congruents are:

`x=2/3pi+K*2pi` and `4/3pi+K*2pi` where `K` is any integer number (positive or negative)

Now the second derivative of `f(x)` is:

`f''(x) = -2sin(x)` which is negative in the first set of points and positive in the second set. So the points:

`x=2/3pi+K*2pi` are all local maxima, and the points
`4/3pi+K*2pi` are all local minima