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Application Of Derivatives

Question
CBSEENMA12035416
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:
sin space 2 straight x comma space space 0 space less than space straight x space less than straight pi


Solution

Let straight f left parenthesis straight x right parenthesis space equals space sin space 2 straight x comma space space space 0 less than straight x less than straight pi
    therefore space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space 2 space cos space 2 straight x space space space space space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space space space space space rightwards double arrow space space space 2 space cos space 2 straight x space equals space 0 space space space space space rightwards double arrow space space cos space 2 straight x space equals space 0 rightwards double arrow space space space space 2 straight x space equals space straight pi over 2 comma space space space fraction numerator 3 straight pi over denominator 2 end fraction space space space space rightwards double arrow space space space space space space straight x space equals space straight pi over 4 comma space space fraction numerator 3 straight pi over denominator 4 end fraction space space space when space straight x space less than space straight pi over 4 space slightly space comma space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space 2 left parenthesis plus ve right parenthesis space equals space plus ve When space straight x greater than straight pi over 4 space slightly comma space straight f space apostrophe left parenthesis straight x right parenthesis space equals 2 left parenthesis negative ve right parenthesis space equals space minus ve space space space space space space
therefore space space space space space at space straight x space equals space straight pi over 4 comma 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 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 straight pi over 4 space and space local space maximum space value space space equals space sin space straight pi over 2 space equals space 1 space space space When space straight x space less than fraction numerator 3 straight pi over denominator 4 end fraction comma space space slightly space straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 left parenthesis negative ve right parenthesis space equals space minus ve space space When space straight x greater than space space fraction numerator 3 straight pi over denominator 4 end fraction space slightly comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 left parenthesis plus ve right parenthesis space equals space plus ve space space space When space straight x greater than fraction numerator 3 straight pi over denominator 4 end fraction comma space space straight f apostrophe left parenthesis straight x right parenthesis space changes space from space minus ve space to space ve therefore 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 3 straight pi over denominator 4 end fraction space and space local space minimum space value space space equals space sin space fraction numerator 3 straight pi over denominator 2 end fraction space equals space minus sin space straight pi over 2 space equals space minus 1

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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