Application of Derivatives

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Question

Find the intervals in which the following function is increasing or decreasing:
$straight f left parenthesis straight x right parenthesis space equals space log space left parenthesis 1 plus straight x right parenthesis space minus space fraction numerator straight x over denominator 1 plus straight x end fraction$

Solution

straight f left parenthesis straight x right parenthesis space equals space log space left parenthesis 1 plus straight x right parenthesis space minus space fraction numerator straight x over denominator 1 plus straight x end fraction

therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator 1 plus straight x end fraction space minus fraction numerator left parenthesis 1 plus straight x right parenthesis. space 1 space minus space straight x. space 1 over denominator left parenthesis 1 plus straight x right parenthesis squared end fraction space equals space fraction numerator 1 over denominator 1 plus straight x end fraction minus fraction numerator 1 over denominator left parenthesis 1 plus straight x right parenthesis squared end fraction space equals space fraction numerator 1 plus straight x minus 1 over denominator left parenthesis 1 plus straight x right parenthesis squared end fraction therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator straight x over denominator left parenthesis 1 plus straight x 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 0 space space space space space space space space rightwards double arrow space space space space space space fraction numerator straight x over denominator left parenthesis 1 plus straight x right parenthesis squared end fraction greater than 0 space space space space space space space rightwards double arrow space space space straight x greater than 0 therefore space space space space straight f left parenthesis straight x right parenthesis space is space increasing space in space left parenthesis 0 comma space infinity right parenthesis

For

straight f left parenthesis straight x right parenthesis space to space be space decreasing comma space space space space straight f apostrophe left parenthesis straight x right parenthesis space less than space 0 space space space space space rightwards double arrow space space space space fraction numerator straight x over denominator left parenthesis 1 plus straight x right parenthesis squared end fraction less than 0 space space space space space rightwards double arrow space space straight x less than 0

therefore space space space straight f left parenthesis straight x right parenthesis space is space decreasing space in space left parenthesis negative infinity comma space 0 right parenthesis