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Application Of Derivatives

Question
CBSEENMA12035420
Wired Faculty App

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
sinx minus cosx comma space space 0 less than straight x less than 2 straight pi






Solution
Let space space straight f left parenthesis straight x right parenthesis space equals space sinx space minus cosx comma space 0 space less than straight x less than 2 straight pi
therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cosx plus sinx straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space rightwards double arrow space space space space cosx plus sinx space equals space 0 space space space space space rightwards double arrow space space space sinx space space equals space minus cosx space space space space rightwards double arrow space space space space tanx space equals space space minus 1 space space space space rightwards double arrow space space tanx space equals negative tan straight pi over 4 space space space space space rightwards double arrow space space space tanx space equals space tan open parentheses straight pi minus straight pi over 4 close parentheses comma space space tan open parentheses 2 straight pi minus straight pi over 4 close parentheses space space space space rightwards double arrow space space space space straight x space equals space fraction numerator 3 straight pi over denominator 4 end fraction comma space fraction numerator 7 straight pi over denominator 4 end fraction
Now,    straight f apostrophe left parenthesis straight x right parenthesis space equals space square root of 2 space open parentheses fraction numerator 1 over denominator square root of 2 end fraction cosx space plus space fraction numerator 1 over denominator square root of 2 end fraction sinx close parentheses space equals space square root of 2 open parentheses sin space straight pi over 4 space cosx space plus space cos space straight pi over 4 space sinx close parentheses

                equals space square root of 2 space sin space open parentheses straight pi over 4 plus straight x close parentheses
When space straight x less than fraction numerator 3 straight pi over denominator 4 end fraction space slightly comma space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space plus ve When space straight x greater than fraction numerator 3 straight pi over denominator 4 end fraction space slightly comma space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus ve therefore space space space at space straight x space equals space fraction numerator 3 straight pi over denominator 4 end fraction comma space space space space straight f apostrophe left parenthesis straight x right parenthesis space changes space from space plus ve space to space minus ve therefore space space space straight f left parenthesis straight x right parenthesis space has space local space maxima space at space straight x space equals space fraction numerator 3 straight pi over denominator 4 end fraction and space local space maximum space value space equals space sin space fraction numerator 3 straight pi over denominator 4 end fraction minus space cos space fraction numerator 3 straight pi over denominator 4 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals sin space open parentheses straight pi minus straight pi over 4 close parentheses minus space cos space open parentheses straight pi minus straight pi over 4 close parentheses space equals space sin straight pi over 4 plus cos space straight pi over 4 equals space fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction equals fraction numerator 2 over denominator square root of 2 end fraction equals square root of 2
When space straight x less than fraction numerator 7 straight pi over denominator 4 end fraction space slightly comma space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus ve When space straight x greater than fraction numerator 7 straight pi over denominator 4 end fraction space slightly comma space straight f apostrophe left parenthesis straight x right parenthesis space equals space plus ve therefore space space space space at space straight x space equals space fraction numerator 7 straight pi over denominator 4 end fraction comma space space straight f apostrophe left parenthesis straight x right parenthesis space space changes space from space minus ve space to space ve therefore space space space space space straight f left parenthesis straight x right parenthesis space has space local space minima space at space straight x space equals space fraction numerator 7 straight pi over denominator 4 end fraction and space local space minimum space value space space equals space sin fraction numerator 7 straight pi over denominator 4 end fraction minus cos space fraction numerator 7 straight pi over denominator 4 end fraction
                                equals space sin space open parentheses 2 straight pi minus straight pi over 4 close parentheses space minus space cos open parentheses 2 straight pi minus straight pi over 4 close parentheses space equals space minus sin space straight pi over 4 space minus space cos space straight pi over 4 equals negative fraction numerator 1 over denominator square root of 2 end fraction minus fraction numerator 1 over denominator square root of 2 end fraction equals space minus fraction numerator 2 over denominator square root of 2 end fraction space equals space minus square root of 2

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