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Application Of Derivatives

Question
CBSEENMA12035432
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:
straight f left parenthesis straight x right parenthesis space equals space straight x square root of 1 minus straight x end root comma space 0 less than straight x less than 1



Solution

  Let space space straight f left parenthesis straight x right parenthesis space equals space straight x square root of 1 minus straight x end root comma space 0 space less than space straight x space less or equal than space 1.
therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space straight x straight d over dx left parenthesis square root of 1 minus straight x end root right parenthesis plus space square root of 1 minus straight x end root. space straight d over dx left parenthesis straight x right parenthesis space space space space space space space space space space space space space space space space equals space straight x. space fraction numerator negative 1 over denominator 2 square root of 1 minus straight x end root end fraction plus square root of 1 minus straight x end root space 1 space equals space fraction numerator negative straight x plus 2 left parenthesis 1 minus straight x right parenthesis over denominator 2 square root of 1 minus straight x end root end fraction equals fraction numerator 2 minus 3 straight x over denominator 2 square root of 1 minus straight x end root end fraction comma space space space space straight x not equal to 1. space space space space space 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 2 minus 3 straight x space equals space 0 space space space space space space space space space rightwards double arrow space straight x space equals space 2 over 3 space space element of space left parenthesis 0 comma space 1 right square bracket therefore space space space space space the space points space where space extremum space may space occur space are left parenthesis 1 right parenthesis space space space straight x space equals space 1 comma space space space the space right space end space point left parenthesis 2 right parenthesis space space space straight x space equals space 2 over 3
When space straight x less than 1 space slightly comma space then space straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator negative ve over denominator plus ve end fraction comma space space equals space minus ve rightwards double arrow space space space space space straight f left parenthesis straight x right parenthesis space has space straight a space local space minima space at space straight x space equals space 1 and space local space minimum space value space space equals space 1 square root of 1 minus 1 end root space equals space 1 space cross times space 0 space equals space 0 Again space when space straight x less than 2 over 3 space slightly comma space then space straight f apostrophe left parenthesis straight x right parenthesis space equals fraction numerator plus ve over denominator plus ve end fraction space equals space plus ve When space straight x greater than 2 over 3 space slightly comma space then space straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator negative ve over denominator plus ve end fraction equals space ve therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space changes space from space plus ve space to space space minus ve space as space straight x space passes space through space 2 over 3 therefore space space space space space straight f left parenthesis straight x right parenthesis space has space straight a space local space maximum space value space at space straight x space equals space 2 over 3 and space this space value space space equals space 2 over 3 square root of 1 minus 2 over 3 end root space equals 2 over 3 cross times fraction numerator 1 over denominator square root of 3 end fraction equals space fraction numerator 2 over denominator 3 square root of 3 end fraction space equals space fraction numerator 2 square root of 3 over denominator 9 end fraction.

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