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# 11.17 - Section 4.3 Derivatives and the shapes of...

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Section 4.3: Derivatives and the shapes of graphs (continued) First Derivative Test : Suppose that c is a critical number of a continuous function f , and suppose that f ( x ) is defined for all x in a neighborhood of c (but not necessarily at c itself). (a) If f changes from positive to negative at c , then f has a local maximum at c . (b) If f changes from negative to positive at c , then f has a local minimum at c . (c) If f is positive on both sides of c or negative on both sides of c , then f has no local maximum or local minimum at c . Proof of (a): Say f ( x ) > 0 for all x in ( c r , c ) and f ( x ) < 0 for all x in ( c , c + r ). Then, by the increasing/decreasing test, f is increasing on [ c r , c ], so that f ( x ) < f ( c ) for all x in ( c r , c ); and likewise, by the increasing/decreasting test, f is decreasing on [ c , c + r ], so that f ( x ) < f ( c ) for all x in ( c , c + r ). Therefore f ( x ) < f ( c ) for all x c in ( c r , c + r ), so f ( x ) f ( c ) for all x in ( c r , c + r ), implying that f has a local maximum at c . (The proofs of (b) and (c) are similar.)

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A common alternative to the First Derivative Test, especially handy when it’s hard to evaluate the derivative of f , is the “test-point method”: Suppose that c is a critical number of a continuous function f , and suppose that we have two test-points d L and d R with d L < c < d R such that c is the only critical point of f in the interval [ d L , d
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11.17 - Section 4.3 Derivatives and the shapes of...

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