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Application Of Derivatives

Question
CBSEENMA12035471
Wired Faculty App

For all real values of x,  the minimum value of fraction numerator 1 minus straight x plus straight x squared over denominator 1 plus straight x plus straight x squared end fraction is

  • 0

  • 1

  • 3

  • 1 third

Solution

D.

1 third Let space space straight y space equals space fraction numerator 1 minus straight x plus straight x squared over denominator 1 plus straight x plus straight x squared end fraction
   space dy over dx space equals space fraction numerator left parenthesis 1 plus straight x plus straight x squared right parenthesis. space begin display style straight d over dx end style left parenthesis 1 minus straight x plus straight x squared right parenthesis space minus space left parenthesis 1 plus straight x plus straight x squared right parenthesis. space begin display style straight d over dx end style left parenthesis 1 plus straight x plus straight x squared right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction

               equals space fraction numerator left parenthesis 1 plus straight x plus straight x squared right parenthesis thin space left parenthesis negative 1 plus 2 straight x right parenthesis minus left parenthesis 1 minus straight x plus straight x squared right parenthesis thin space left parenthesis 1 plus 2 straight x right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction equals fraction numerator negative 1 plus 2 straight x minus straight x plus 2 straight x squared minus straight x squared plus 2 straight x cubed minus 1 minus 2 straight x plus straight x plus 2 straight x squared minus straight x squared minus 2 straight x cubed over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction equals space fraction numerator negative 2 plus 2 straight x squared over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis end fraction space equals space fraction numerator 2 left parenthesis straight x squared minus 1 right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction space equals space fraction numerator 2 left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction
dy over dx space equals 0 space space space space rightwards double arrow space space space space fraction numerator 2 left parenthesis straight x squared minus 1 right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis end fraction space equals space 0 space space space space rightwards double arrow space space space straight x squared minus 1 space equals space 0 space space space space rightwards double arrow space space space straight x space equals space minus 1 comma space space 1 When space straight x less than negative 1 space slightly comma space space space dy over dx space equals space plus ve When space straight x space greater than negative 1 space slightly comma space dy over dx space equals space minus ve therefore space space space space space at space straight x space equals space minus 1 comma space space space dy over dx space changes space from space plus ve space to space minus ve therefore space space space space space space straight y space is space max comma space when space straight x space equals space minus 1 we space are space not space increased space in space this space case. When space straight x less than 1 space slightly comma space dy over dx space equals negative ve When space straight x greater than 1 space sightl comma space dy over straight d equals plus ve therefore space space at space straight x space equals space 1 comma space space dy over dx space changes space from space minus ve space to space plus ve therefore space space space space space space space straight y space is space minimum space when space straight x space equals space 1 and space minimum space value space space equals space fraction numerator 1 minus 1 plus 1 over denominator 1 plus 1 plus 1 end fraction space equals 1 third therefore space space space left parenthesis straight D right parenthesis space is space correct space answer.

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