Calc I Honors
3.1 – 3.2 Review Notes
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Definition of Extrema
Let f be defined on an interval I containing c.
1. f(c) is the maximum of f on I if f(c) > f(x) for all x on I.
2. f(c) is the minimum of f on I if f(c) < f(x) for all x on I.
The maximum and minimum of a function on an interval are the extreme values, or
extrema, of the function on the interval. The maximum and minimum
function on an interval are also called the absolute maximum and absolute minimum on
the interval, respectfully.
Definition of a Critical Number
Critical numbers are numbers you check to locate any extrema.
Let f be defined at c. If f ‘(c) = 0 or if f is undefined at c, then c is a critical number
Theorem: Relative Extrema Occur only at Critical Numbers
If f has a relative minimum or relative maximum at x = c, then c is a critical number of f.
Guidelines for Finding Extrema on a Closed Interval
To find the extrema of a continuous function f on
a closed interval [a, b], use the following steps.
1) Find critical numbers of f on (a, b).
2) Evaluate f at each critical number in (a, b).
3) Evaluate f at each endpoint of [a, b].
4) The least of this values is the minimum. The
greatest of these values is the maximum.
3.2 Rolle’s Theorem
Rolle’s Theorem
Let f be continuous on the closed interval [a, b] and differentiable on the open interval
(a,b). If f(a) = f(b), then there is at least one number c in (a,b) such that f’(c) = 0.
Put this in your own words:
Examples:
MEAN VALUE THEOREM
If f is continuous on the closed interval [a, b] and differentiable on the open interval
(a,b), then there exists a number c in (a,b) such that:
f’(c) =
f (b)  f ( a )
ba