Extreme value theorem

What is the extreme value theorem?

If f is continuous on a close interval [a,b], then f has both an absolute minimum and an absolute maximum at some numbers c and d in the interval.

What is Fermat’s theorem?

If f has a local maximum or local minimum at c, and if f’(c ) exists, then f’(c )=0

What is the procedure for finding the absolute extrema of a continuous function on a closed interval?

1. find critical numbers of f in [a,b]

2. add a and b to the list of numbers in step 1

3. evaluate the original function at each of the numbers from steps 1 and 2. the largest is the absolute max, the smallest is the absolute min

if there is only one critical number, how do you know the absolute min/max?

its the relative and absolute max/min because it’s the only critical number

Practice

Answer