Section 4.1 Maximum and minimum values
Definition. Let cbe a number in the domain Dof a function f. Then f(c) is the
•absolute maximum value of fon D, if f(c)≥f(x) for all xin D.
•absolute minimum value of fon D, if f(c)≤f(x) for all xin D.
Definition. A number f(c) is a
•local maximum value, if f(c)≥f(x) when xis near c.
•local minimum value, if f(c)≤f(x) when xis near c.
Example 1. Sketch the graph of the function fthat is continuous on [0,3] and has the absolute maximum at
0, absolute minimum at 3, local minimum at 1, local maximum at 2.
The extreme value theorem. If fis continuous on a closed interval [a,b], then fattains an absolute
maximum value f(c) and an absolute minimum value f(d) at some numbers cand din [a, b].
Fermat’s theorem. If fhas a local maximum or minimum at c, and if f′(c) exists, then f′(c) = 0
Definition. Acritical number of a function fis a number cin the domain of fsuch that either
f′(c) = 0 or f′(c) does not exist.
1
