Sponsor Area

Application Of Derivatives

Question
CBSEENMA12035395

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
straight f left parenthesis straight x right parenthesis space equals space sin space straight x plus space cosx comma space space straight x space space element of space space left square bracket 0 comma space straight pi right square bracket

Solution

Here straight f left parenthesis straight x right parenthesis space equals space sin space straight x plus space cos space straight x
It is differentiable for all x in open square brackets 0 comma space space straight pi close square brackets
therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cosx minus space sinx space
space space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space gives space cosx minus space sinx space equals space 0
therefore space space space sin space straight x space equals space cosx space space space space or space space space space space tan space straight x space equals space 1 space space space space space space space rightwards double arrow space space space space straight x space equals space straight pi over 4 space element of space space space open square brackets 0 comma space straight pi close square brackets
space space space space space space straight f open parentheses straight pi over 4 close parentheses space equals space sin space straight pi over 4 plus space cos space straight pi over 4 space equals space fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction space equals space fraction numerator 2 over denominator square root of 2 end fraction space equals space square root of 2
straight f left parenthesis 0 right parenthesis space equals space sin space 0 plus space cos space 0 space equals space 0 plus 1 space equals space 1
straight f left parenthesis straight pi right parenthesis space equals space space sin space straight pi space plus space cos space straight pi space equals space 0 minus 1 space equals space minus
therefore space space space absolute space max. space value space space equals space square root of 2 space and space absolute space min. space value space space equals space minus 1.