3.1 Maximum and Minimum Values, Exams of Calculus

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3.1 Maximum and Minimum Values

The Extreme Value Theorem

So, if we have a continuous function on an interval [ a,b ] then we are guaranteed to have both an absolute maximum and an absolute minimum for the function somewhere in the interval. The theorem doesn’t tell us where they will occur or if they will occur more than once, but at least it tells us that they do exist somewhere. Sometimes, all that we need to know is that they do exist. Existence Only!!

The converse however is not necessarily true!!

Consider the absolute value of x at 0.

Consider y=x

at 0

Definition: A critical number of a function f is a number"c" in the domain of f such that either f ' (c) = 0 or f ' (c) does not exist.

If f has a local max or min at c, then c is a critical value of f.

To Find the Absolute Max and Min Values on [a,b]

1. Find all critical values of the function. (Take the derivative and find zeroes and undefined points.)
2. Find the values of f with the critical points and endpoints plugged in. Do this clearly!!
3. The largest value is the Absolute Max and the smallest Value is the Absolute Min. These values are yvalues (not x!).

Attachments extreme value example.mp