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Application Of Derivatives

Question
CBSEENMA12035292
Wired Faculty App

Determine the values of x for which the function straight f left parenthesis straight x right parenthesis space equals fraction numerator straight x over denominator straight x squared plus 1 end fraction is increasing and for which it is decreasing.

Solution

Here straight f left parenthesis straight x right parenthesis space equals fraction numerator straight x over denominator straight x squared plus 1 end fraction
therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals fraction numerator left parenthesis straight x squared plus 1 right parenthesis. space 1 minus straight x. space 2 straight x over denominator left parenthesis straight x squared plus 1 right parenthesis squared end fraction space equals space fraction numerator 1 minus straight x squared over denominator left parenthesis straight x squared plus 1 right parenthesis squared end fraction
For straight f left parenthesis straight x right parenthesis to be increasing,  straight f apostrophe left parenthesis straight x right parenthesis space greater than space 0
therefore space space space space space fraction numerator 1 minus straight x squared over denominator left parenthesis straight x squared plus 1 right parenthesis squared end fraction greater than 0 space space space space space rightwards double arrow space space 1 minus straight x squared greater than 0 space space space space space rightwards double arrow space space space 1 greater than straight x squared rightwards double arrow space space space space space straight x less than 1 space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space open vertical bar straight x squared close vertical bar space less than space space 1 space space space space space space space space rightwards double arrow space space space space open vertical bar straight x close vertical bar space less than space 1 rightwards double arrow space space space space space 1 space less than space straight x space space less than space 1 therefore space space space space straight f left parenthesis straight x right parenthesis space is space increasing space for space minus 1 less than straight x less than 1 therefore space space space space straight f left parenthesis straight x right parenthesis space is space increasing space when space straight x space element of space left parenthesis negative 1 comma space 1 right parenthesis Again space straight f left parenthesis straight x right parenthesis space is space decreasing space when space straight f apostrophe left parenthesis straight x right parenthesis space less than space 0 space therefore space space space space space fraction numerator 1 minus straight x squared over denominator left parenthesis straight x squared plus 1 right parenthesis squared end fraction less than 0 space space space space space space space space rightwards double arrow space space space space space space 1 minus straight x squared less than 0 space space space space space space space space space rightwards double arrow space space space space 1 less than straight x squared
rightwards double arrow space space space space straight x squared greater than 1 space space space space space rightwards double arrow space space space space straight x squared greater than 1 space space space space space space space rightwards double arrow space space space space open vertical bar straight x close vertical bar squared greater than 1 space space space space space space space rightwards double arrow space space space space space open vertical bar straight x close vertical bar greater than 1 rightwards double arrow space space space either space straight x less than negative 1 space space space space space or space space space space space straight x greater than 1 therefore space space space space space straight f left parenthesis straight x right parenthesis space is space decreasing space for space straight x less than negative 1 space space space space or space space space space space straight x greater than 1 therefore space space space space straight f left parenthesis straight x right parenthesis space is space decreasing space when space straight x comma space element of space space space space left parenthesis negative infinity comma space space minus 1 right parenthesis space union space space space space space left parenthesis 1 comma space space space infinity right parenthesis

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