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Chap4_Sec1 - MAXIMUM MINIMUM VALUES Example 4(4.1 You can...

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MAXIMUM & MINIMUM VALUES You can see that f (1) = 5 is a local maximum, whereas the absolute maximum is f (-1) = 37. square4 This absolute maximum is not a local maximum because it occurs at an endpoint. Also, f (0) = 0 is a local minimum and f (3) = -27 is both a local and an absolute minimum. square4 Note that f has neither a local nor an absolute maximum at x = 4. Example 4 (4.1)
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If f ( x ) = x 3 , then f ’ ( x ) = 3 x 2 , so f ’ (0) = 0. square4 However, f has no maximum or minimum at 0—as you can see from the graph. square4 Alternatively, observe that x 3 > 0 for x > 0 but x 3 < 0 for x < 0. The fact that f ’ (0) = 0 simply means that the curve y = x 3 has a horizontal tangent at (0, 0). square4 Instead of having a maximum or minimum at (0, 0), the curve crosses its horizontal tangent there. Example 5 (4.1) FERMAT’S THEOREM
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The function f ( x ) = | x | has its (local and absolute) minimum value at 0. square4 However, that value can’t be found by setting f ’ ( x ) = 0. square4 This is because— f ’ (0) does not exist.
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