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# notes9 - CM221A ANALYSIS I NOTES ON WEEK 9 FINDING MAXIMAL...

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CM221A ANALYSIS I NOTES ON WEEK 9 FINDING MAXIMAL AND MINIMAL VALUES Deﬁnition. Let f be a function deﬁned on an interval ( a,b ). We say that f has a local maximum at a point c ( a,b ) if there exists ε > 0 such that f ( c ) > f ( x ) for all x ( c - ε,c + ε ). Similarly, f has a local minimum at c ( a,b ) if there exists ε > 0 such that f ( c ) 6 f ( x ) for all x ( c - ε,c + ε ). Theorem. Let f be a diﬀerentiable function on the interval ( a,b ). If f has a local maximum or a local minimum at c ( a,b ) then f 0 ( c ) = 0. Proof. If f has a local maximum at c then there exists ε > 0 such that f ( x ) - f ( c ) x - c > 0 for all x ( c - ε,c ) and f ( x ) - f ( c ) x - c 6 0 for all x ( c,c + ε ). Therefore, in the deﬁnition of the derivative, the left limit is nonnegative and the right limit is nonpositive. Since f is diﬀerentiable, both these limits exist and coincide. This implies that they are equal to zero. The corresponding result for a local minimum is obtained in a similar way (or by applying the local maximum result to the function g ( x ) = - f ( x )).

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notes9 - CM221A ANALYSIS I NOTES ON WEEK 9 FINDING MAXIMAL...

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