Absolute and Local Extrema

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Differentiation Review

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Absolute and Local Extrema

Definitions
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Critical Numbers
Steps to Find Absolute Extrema

The Mean Value and other Theorems

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$f$ vs. $f'$

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The Fundamental Theorem of Calculus

Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
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Critical numbers

The Extreme Value Theorem tells us that the minimum and maximum of a continuous function on a closed interval have to be somewhere. But where should we look?

It turns out that extreme values can occur at the endpoints of the intervals, or at the "top of a hill" or the "bottom of a valley" of the graph of a function. How can we find these hills and valleys? They can occur only at $x$ -values where the derivative either equals zero (where the tangent line has slope of zero), or does not exist (where there may be a cusp).

Fermat's Theorem: Suppose that

$a \lt c \lt b$ . If a function

$f$ is defined on the interval

$(a,b)$ , and it has a maximum or a minimum at

$c$ , then either

$f'$ doesn't exist at

$c$ or

$f'(c)=0$ .

Such $x$ -values $c$ are called critical numbers.

Critical numbers: A critical number of a function

$f$ is a number

$c$ in the domain of

$f$ such that

$f'(c)=0$ or

$f'(c)$ is not defined.

What we are saying, then, is that absolute maxima or minima can only occur at critical numbers and endpoints of intervals.

Equivalently, if

$f'(c)$ exists and is not zero, then

$f(c)$ is neither a maximum nor a minimum.

The process of finding such maxima and minima is not difficult in practice: