Optimization Math Example 4

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Example 4

medium

Find the absolute maximum and minimum of

f(x) = x^3 - 6x^2 + 9x + 1

[0, 4]

Solution

1
$f'(x) = 3x^2 - 12x + 9 = 3(x-1)(x-3)$ . Critical points in $[0,4]$ : $x=1$ and $x=3$ .
2
Evaluate $f$ at critical points and endpoints: $f(0)=1$ , $f(1)=5$ , $f(3)=1$ , $f(4)=5$ .
3
Absolute maximum: $5$ (at $x=1$ and $x=4$ ). Absolute minimum: $1$ (at $x=0$ and $x=3$ ).

Answer

Absolute maximum

= 5

; absolute minimum

= 1

For absolute extrema on a closed interval, check all critical points inside the interval AND both endpoints. The largest and smallest values among these candidates are the absolute maximum and minimum.

About Optimization

The process of using derivatives to systematically find maximum or minimum values of a function over a domain.

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More Optimization Examples

Example 1 easy

Find the local maximum and minimum values of [formula].

Example 2 hard

A farmer has 200 m of fencing to enclose a rectangular field. What dimensions maximize the enclosed

Example 3 easy

Find the critical points of [formula] and classify them.

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Derivative