Finding Absolute Extrema

Finding absolute extrema simply means finding the single highest and single lowest value a function reaches on some stretch $[a, b]$. All we're really doing is making a short list of suspects and checking each one. The formula is just that list: $f(a)$ and $f(b)$ (the values at the two ends) and $f(c)$ for each critical point $c$ (the values at the flat spots and corners in between). Whichever number is biggest is the max, and whichever is smallest is the min. It really is that simple.

Last lesson gave us the two pieces we need. The Extreme Value Theorem says a continuous function on $[a, b]$ definitely has an absolute max and min. And Fermat's theorem says that if one of them is inside the interval, it's at a peak or valley, so at a critical point. The only other place it could be is at an end, where the graph gets cut off and doesn't need to be flat. Put those together and there's nowhere else to look.

The closed interval method:

1. Check $f$ is continuous on $[a, b]$.
2. Find the critical points inside $(a, b)$, and ignore any outside it.
3. Evaluate $f$ at those critical points and at $a$ and $b$.
4. The largest value is the absolute max, and the smallest is the absolute min.

Notice you don't need to figure out which critical points are peaks and which are valleys. You're just comparing a handful of numbers.

Any "best setting on a dial" question, like the fastest safe speed for a delivery route or the dose with the best effect within a safe range, is a closed interval problem.