121616949-math.109 - that on both sides of x 2 the values...

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5.1 Maxima and Minima 95 . Figure 5.2 No maximum or minimum even though the derivative is zero. we compute the value of f ( a ) for x 1 < a < x 2 , and that f ( a ) < f ( x 2 ). What can we say about the graph between a and x 2 ? Could there be a point ( b, f ( b )), a < b < x 2 with f ( b ) > f ( x 2 )? No: if there were, the graph would go up from ( a, f ( a )) to ( b, f ( b )) then down to ( x 2 , f ( x 2 )) and somewhere in between would have a local maximum point. But at that local maximum point the derivative of f would be zero or nonexistent, yet we already know that the derivative is zero or nonexistent only at x 1 , x 2 , and x 3 . The upshot is that one computation tells us that ( x 2 , f ( x 2 )) has the largest y coordinate of any point on the graph near x 2 and to the left of x 2 . We can perform the same test on the right. If we find
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Unformatted text preview: that on both sides of x 2 the values are smaller, then there must be a local maximum at ( x 2 , f ( x 2 )); if we find that on both sides of x 2 the values are larger, then there must be a local minimum at ( x 2 , f ( x 2 )); if we find one of each, then there is neither a local maximum or minimum at x 2 . x 1 a b x 2 x 3 • • • • Figure 5.3 Testing for a maximum or minimum. It is not always easy to compute the value of a function at a particular point. The task is made easier by the availability of calculators and computers, but they have their own drawbacks—they do not always allow us to distinguish between values that are very...
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