Answer:
Absolute minimum: x = -π / 6
Absolute maximum: x = π
Step-by-step explanation:
The candidates for the absolute maximum and minimum are the endpoints and the critical points of the function.
First, we evaluate the function at the endpoints.
At x = -π, we have


At x = π, we have


Next, we find the critical points and evaluate the function at them.
The critical points = are points where the first derivative of the function are zero.
Taking the first derivative of the function gives


Now the critical points are where df(x)/dx =0; therefore, we solve

solving for x gives


on the interval [−π,π].
Now, we evaluate the function at the critical points.
At x = -π/ 6, we have


At x = -5π/6, we have


Hence, our candidates for absolute extrema are

Looking at the above we see that the absolute maximum occurs at x = π and the absolute minimum x = -π/6.
Hence,
Absolute maximum: x = π
Absolute minimum: x = -π / 6