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Section 4.1

Maxima and Minima

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a. Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x = c.

Absolute maximum at x = c

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Absolute maximum: q and sLocal maximum: q and s

Absolute minimum: pLocal minimum: p and r

If f has a local maximum or minimum at cand f’(c) exists, then f’(c) = 0.

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Note the converse of Theorem is not necessarily true.

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X(2ln x + 1) = 0; x = 0 not in domain

or 2ln x + 1 = 0 ln x = -1/2 x = e-1/2 ≈ 0.61

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f(-2) = 32 absolute maximum

f(3/2) = -27/16 absolute minimum

a. 22 0 2 3 0

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x x

x x critical values

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b.

g(-1) = 3 absolute maximum and g(0) = g(2) = 0 absolute minimum

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SOLUTION