Application of Derivatives

Find the intervals in which the function f is given by is (i) increasing (ii) decreasing - Wired Faculty

Question

Find the intervals in which the function f is given by
$straight f left parenthesis straight x right parenthesis space equals space fraction numerator 4 space sin space straight x space minus space 2 straight x minus space straight x space cosx over denominator 2 plus cos space straight x end fraction$
is (i) increasing (ii) decreasing

Solution

Here $straight f left parenthesis straight x right parenthesis space equals space fraction numerator 4 space sin space straight x space minus space 2 straight x minus space straight x space cosx over denominator 2 plus cos space straight x end fraction space equals space fraction numerator 4 sinx space minus space straight x left parenthesis 2 plus cosx right parenthesis over denominator 2 plus cosx end fraction space equals fraction numerator 4 space sinx over denominator 2 plus cosx end fraction minus straight x$
$therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator left parenthesis 2 plus cos space straight x right parenthesis thin space left parenthesis 4 space cos space straight x right parenthesis space minus space left parenthesis 4 space sin space straight x right parenthesis space left parenthesis negative sin space straight x right parenthesis over denominator left parenthesis 2 plus cosx right parenthesis squared end fraction minus 1$
   $equals space fraction numerator 8 space cosx space plus space 4 space cos squared straight x space plus space 4 space sin squared straight x over denominator left parenthesis 2 plus cos space straight x right parenthesis squared end fraction minus 1 space equals space fraction numerator 8 space cosx space plus space 4 left parenthesis cos squared straight x plus sin to the power of 2 straight x end exponent right parenthesis over denominator left parenthesis 2 plus cos space straight x right parenthesis squared end fraction minus 1 equals space fraction numerator 8 space cosx space plus space 4 over denominator left parenthesis 2 plus cosx right parenthesis squared end fraction minus 1 space equals space fraction numerator 8 space cos space straight x plus space 4 space minus space left parenthesis 2 plus space cosx right parenthesis squared over denominator left parenthesis 2 plus cosx right parenthesis squared end fraction therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator 8 cosx plus 4 minus 4 minus cos squared straight x minus 4 cosx over denominator left parenthesis 2 plus cosx right parenthesis squared end fraction space equals space fraction numerator 4 space cosx minus cos squared straight x over denominator left parenthesis 2 space plus cosx right parenthesis squared end fraction therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator cosx space left parenthesis 4 minus cosx right parenthesis over denominator left parenthesis 2 plus cosx right parenthesis squared end fraction$
(i) For f(x) to be increasing.
   $straight f apostrophe left parenthesis straight x right parenthesis space greater than space 0 space space space space space space rightwards double arrow space space space space space space fraction numerator cosx left parenthesis 4 minus cosx right parenthesis over denominator left parenthesis 2 plus cosx right parenthesis squared end fraction greater than 0$
$rightwards double arrow space space space cosx space greater than space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space left parenthesis 2 plus cosx right parenthesis squared greater than 0 comma space space space 4 minus cosx greater than 0 space as space minus straight i less or equal than cosx less or equal than 1 close square brackets rightwards double arrow space space space 0 space less than space straight x thin space less than straight pi over 2 space space space or space space space space fraction numerator 3 straight pi over denominator 2 end fraction less than straight x less than 2 straight pi$
(ii) For f(x) to be decreasing,
$straight f apostrophe left parenthesis straight x right parenthesis thin space less than 0 space space space space space space space space space space rightwards double arrow space space space space space space fraction numerator cosx space left parenthesis 4 minus cosx right parenthesis over denominator left parenthesis 2 plus cosx right parenthesis squared end fraction space less than space 0 space space space space space rightwards double arrow space space space space space cosx less than 0 therefore space space space space space straight pi over 2 less than straight x less than fraction numerator 3 straight pi over denominator 2 end fraction$

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