Blog for 10/17

well, this week we learned how to find extremas

Definition of Extrema
Let f be defined on an interval I containing c. f(c) is the minimum of f on I if f(c) < f(c) for all x in I
f(c) is the minimum of f on I if f(c) > f(c) for all x in I
The minimum and max of a function on an interval are the extreme values or extrema of the function on the interval. The minimum and max of a function on an interval are also called the absolute min and absolute max on the interval, respectively.

Definition of Relative Extrema
If there is an open interval containing c on which f(c) is a maximum, then f(c) is called a relative maximum of f.
If there is an open interval containing c on which f(c) is a minimum, then f(c) is called a relative minimum of f.

Definition of Critical Number
Let f be defined at c. If f '(c) = 0 or if f ' is undefined at c, then c is a critical number of f.

Example:

Find the extrema of f(x) = 3x3 - 4x4 on the interval [-1, 2].

Find the critical numbers
f(x) = 3x3 - 4x4   {Problem}
f '(x) = 9x2 - 16x3   {Differentiate}
9x2 - 16x3 = 0   {Set f '(x) = 0}
x(9x - 16) = 0   {Factor}
x = 0, 16/9   {solve to get the critical numbers}