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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 )).
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online