3 Applications of Differentiation

3.1 Maximum and Minimum Values

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Maximum and Minimum Values

Some of the most important applications of calculus are optimization problems, which find the optimal way of doing something. Meaning: find the maximum or minimum values of some function.

So what are maximum and minimum values?

Example: highest point on the graph of the function f shown is the point (3, 5) so the largest value off is f (3) = 5. Likewise, the smallest value is f (6) = 2.

Figure 1

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Absolute Maximum and Minimum

The maximum and minimum values of f are called extreme values of f.

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Local Maximum and Minimum

Something is true near c means that it is true on some open interval containing c.

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Example 1

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Figure 2 shows the graph of a function f with several extrema:

At a : absolute minimum is f(a)

At b : local maximum is f(b)At c : local minimum is f(c)At d : absolute (also local) maximum is f(d)

At e : local minimum is f(e)

Figure 2

Abs min f (a), abs max f (d)loc min f (c) , f(e), loc max f (b), f (d)

Example 2

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Practice! Define all local and absolute extrema of the graph below

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Maximum and Minimum Values

The following theorem gives conditions under which a

function is guaranteed to have extreme values.

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Maximum and Minimum Values

The Extreme Value Theorem is illustrated in Figure 7.

Note that an extreme value can be taken on more than

once.

Figure 7

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Maximum and Minimum Values

The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values. We start by looking for local extreme values.

Figure 10 shows the graph of a function f with a local maximum at c and a local minimum at d.

Figure 10

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Counter examples:

f’ doesn’t exist but there is a local min or max

f’ exists but there is no local min or max

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Maximum and Minimum Values

To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local or it occurs at an endpoint of the interval.

Thus the following three-step procedure always works.