Question

What are the extrema of `f(x)= x+sinx in [-pi,pi]` ?

Answer 1

Absolute maximum at `pi` and an absolute minimum at `-pi`.

Explanation:

For absolute extrema, we need to find the critical numbers by setting the derivative of the equation equal to 0 and plugging them into the function, and then plug in the endpoints of the interval into the original equation.

1. Finding the critical numbers
   `f'(x) = 1 + cosx`
   `1 + cosx = 0`
   `cosx = -1`
   `x=pi` <-- critical number
   `f(pi)=pi`

2. Plug in the endpoints
   `f(-pi)=-pi`
   `f(pi)=pi`

Comparing the three, we find that we have an absolute maximum at `pi` and an absolute minimum at `-pi`.