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Application Of Derivatives

Question
CBSEENMA12035344

Find the intervals in which the following function is increasing or decreasing:
f (x) = sinx – cosx, 0 < x < 2straight pi

Solution

f(x) = sinx – cosx
therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cosx plus sinx space equals space square root of 2 space open square brackets fraction numerator 1 over denominator square root of 2 end fraction cosx plus fraction numerator 1 over denominator square root of 2 end fraction sinx close square brackets
space space space space space space space space space space space space space space space equals space square root of 2 open square brackets sin straight pi over 4 cosx plus cos space straight pi over 2 space sin space straight x close square brackets
therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space square root of 2 space sin space open parentheses straight x plus straight pi over 4 close parentheses
For straight f left parenthesis straight x right parenthesis to be increasing,   straight f apostrophe left parenthesis straight x right parenthesis greater than 0
therefore space space space square root of 2 space sin space open parentheses straight x plus straight pi over 4 close parentheses greater than 0 space space space space space space space space space rightwards double arrow space space space space sin space open parentheses straight x plus straight pi over 4 close parentheses greater than 0
rightwards double arrow space space space 0 space less than space straight x space plus space straight pi over 4 less than space straight pi space space space space space space space rightwards double arrow space space space minus straight pi over 4 less than straight x less than fraction numerator 3 straight pi over denominator 4 end fraction space space space space space space space space space space space space space space space space space left parenthesis because space 0 less than straight x less than 2 straight pi right parenthesis
therefore space space space space space straight x space element of space space open parentheses 0 comma space fraction numerator 3 straight pi over denominator 4 end fraction close parentheses
For straight f left parenthesis straight x right parenthesis to be decreasing,   straight f apostrophe left parenthesis straight x right parenthesis space less than space 0
therefore space space space square root of 2 space sin space open parentheses straight x plus straight pi over 4 close parentheses space less than 0 space space space space space space space rightwards double arrow space space space sin space open parentheses straight x plus straight pi over 4 close parentheses space less than 0
rightwards double arrow space space straight pi less than space straight x space plus space straight pi over 4 less than space 2 straight pi space space space space space space space space space space space space rightwards double arrow space space space fraction numerator 3 straight pi over denominator 4 end fraction less than straight x less than fraction numerator 7 straight pi over denominator 4 end fraction
therefore space space straight x space space element of space space open parentheses fraction numerator 3 straight pi over denominator 4 end fraction comma space space fraction numerator 7 straight pi over denominator 4 end fraction close parentheses