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Application Of Derivatives

Question
CBSEENMA12035405
Wired Faculty App

Find the absolute maximum and minimum values of the function f given by
 f (x) = cos2x + sinx,  x ∊ [0, straight pi].

Solution
straight f left parenthesis straight x right parenthesis space equals cos squared straight x plus space sinx
It is differentiable for all x in open square brackets 0 comma space straight pi close square brackets.
straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 space cos space straight x left parenthesis negative sin space straight x right parenthesis space space plus space space cos space straight x space space equals space cosx space left parenthesis 1 minus 2 space sin space straight x right parenthesis
straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space gives space cosx left parenthesis 1 minus 2 space sinx right parenthesis space equals space 0
therefore space space space space space cosx space equals space 0 therefore space space space space space straight x space equals space straight pi over 2 or space space space 1 minus 2 space sinx space space equals space 0 therefore space space space space space sin space straight x space equals space 1 half therefore space space space sin space straight x space equals space sin space straight pi over 6 comma space space sin space fraction numerator 5 straight pi over denominator 6 end fraction therefore space space space space space space space straight x space equals space straight pi over 6 comma space space fraction numerator 5 straight pi over denominator 6 end fraction therefore space space space space space space space straight x space equals space straight pi over 3 comma space straight pi over 2 comma space fraction numerator 5 space straight pi over denominator 6 end fraction
  At straight x equals straight pi over 6 comma space space straight f left parenthesis straight x right parenthesis space equals space cos squared straight pi over 6 plus sin straight pi over 6 space equals open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses squared plus 1 half space equals space 3 over 4 plus 1 half space equals space 5 over 4
 
At space space straight x space equals space fraction numerator 5 straight pi over denominator 6 end fraction comma space space space space straight f left parenthesis straight x right parenthesis space equals space cos squared fraction numerator 5 straight pi over denominator 6 end fraction plus sin space fraction numerator 5 straight pi over denominator 6 end fraction space equals space cos squared open parentheses straight pi minus straight pi over 6 close parentheses plus sin space open parentheses straight pi minus straight pi over 6 close parentheses space space space space space space space space space space space equals space cos squared straight pi over 6 plus sin space straight pi over 6 space equals space 5 over 4 space At space straight x space equals space straight pi over 2 comma space space space straight f left parenthesis straight x right parenthesis space equals space cos squared straight pi over 2 plus sin space straight pi over 2 space equals space 0 plus 1 space equals space 1 At space space space straight x space equals space 0 comma space space space straight f left parenthesis straight x right parenthesis space equals space cos squared straight pi over 4 space plus space sin space straight pi space equals space left parenthesis negative 1 right parenthesis squared plus 0 space equals space 1 plus 0 space equals space 1 therefore space absolute space max space space value space space equals space 5 over 4 space and space aboslute space minimum space value space space equals space 1
 

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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