The Extreme Value Theorem is an extension of a function's critical points. If a function is continuous over a closed interval, there are maximum and minimum values of that function. To find the 'extreme' values, we must first make certain the function is continuous over the interval, then find the critical points. The final step is to find the minimum and maximum values of the critical points AND the start and end points of the interval to determine the maximum and minimum values. In this example, we find the 'extreme values' (minimum and maximum) values of:
$f\left(x\right)=sinx+cosx$
on the interval [0,2Π]
Extreme Vaue Theorem