critical number: a number$\ c$ in the domain of a function$\ f$ such that:
Intros
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 How to describe graphs of functions?
 How to describe graphs of functions?
$\bullet$ local maximum
$\bullet$ local minimum
$\bullet$ critical number
 How to describe graphs of functions?
state the:
$\bullet$ absolute maximum
$\bullet$ absolute minimum

on the interval,
$1\leq x\leq 12\,$state the:
$\bullet$ absolute maximum
$\bullet$ absolute minimum
Examples
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 Find the critical numbers of the function:
 First Derivative Test: a test to determine whether or not$\ f$ has a local maximum or minimum at a critical number
First Derivative Test
local maximum
local minimum
no maximum or minimum
no maximum or minimum
 $f(x)=3x^{5}15x^{4}+25x^{3}15x^{2}+5$
 The Closed Interval Method
To find the absolute maximum and minimum values of a continuous function $f$ on a closed interval [a, b]:
1.Find the values of $f$ at the critical numbers of $f$ in (a, b).
2.Find the values of $f$ at the leftendpoint and rightendpoint of the interval
3.Compare all values from steps 1 and 2: the largest$\$is the absolute maximum value; the smallest$\$is the absolute minimum value.  Find the absolute maximum and minimum values of the function:
$f(x)=3x^{5}15x^{4}+25x^315x^2+5$$\frac{1}{2}\leq x\leq\frac{1}{2}$