Extrema

Definition of an Extrema

The extrema of a function f are the values where f is either a maximum or a minimum.  More rigorously, we have

 Definition of Extrema Let f  be a function defined on the interval (a,b) containing the point c.  Then   f  has a minimum at c if f (c) <  f (x) for all x in (a,b). f  has a maximum at c if f (c) >  f (x) for all x in (a,b).

The Extreme Value Theorem

A continuous function on a closed interval must always have a maximum and a minimum.

 The Extreme Value Theorem If f  is a continuous function on [a,b], then f  has both a maximum and a minimum.

Note:  The requirement that f is continuous is essential.

Definition of a Relative Extrema

Often we are considered how a point compares only with its neighbors.  If a function evaluated at a point is the largest among all nearby function values, then we say that the function has a relative maximum.  Similarly, if the function evaluated at a point is the largest among all nearby function values, then we say that the function has a relative minimum.

 Definition of a Relative Extrema Let f  be a function defined on the interval (a,b) containing the point c.  Then f  has a relative maximum at c if            f (c) >  f (x)  for all x in some interval (u,v) containing c. f  has a relative minimum at c if            f (c) <  f (x)  for all x in some interval (u,v) containing c.

Note:  Any Extrema is also a relative extrema, but the converse does not hold.

Definition of a Critical Number

A value c is called a critical number of a function f  if either

1. f '(c) = 0 or

2. f '(c) does not exist.

 Theorem:   If f  has a relative extrema at c, then c is a critical number for f .

Proof:
(For the case of a relative maximum)

Now notice that since c is a relative maximum, the numerator is negative.  Since the denominator takes on negative values for x < c and positive values for x > c, the derivative is both positive and negative.  This can only occur if it is zero or does not exist.

From the two theorems, the extrema of a closed interval can only occur at either a critical point or an end point.  So to find the extrema, set the derivative equal to 0, and solve.  Plug the solutions and the endpoints back into the original equation and the largest y value will be the maximum, while the smallest will be the minimum.

Example

Find the relative extrama of

f (x) = x3 - 3x + 4

on the interval [0,2]

Solution

f '(x) = 3x2 - 3

which has zeros at -1 and 1.  Now find y,

 x f (x) 0 4 1 2 2 6

Hence the maximum is 6 and occurs at x = 2, while the minimum is 2 and occurs at x = 1.