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Find the greatest and least values

  1. Jun 10, 2013 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    Find the greatest and least values of the function [itex]f(x)=(sin^{-1}x)^3 +(cos^{-1}x)^3[/itex]

    2. Relevant equations

    3. The attempt at a solution
    Setting f'(x)=0 and solving I get [itex]|sin^{-1}x|=|cos^{-1}x|[/itex]
     
  2. jcsd
  3. Jun 10, 2013 #2

    verty

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    What does ##sin^{-1}{x}## mean? The remainder of the question is a test of your intuition. I hope no one gives too much help.
     
  4. Jun 10, 2013 #3

    Ray Vickson

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    The notation ##\sin^{-1} x## is equally likely to mean ##1/ \sin\, x## or ##\arcsin\, x##. After all, the notation ##\sin^n x## is taken to mean ##(\sin \, x)^n## whenever ##n \neq -1##! So, which do you mean?
     
  5. Jun 10, 2013 #4

    utkarshakash

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    I mean arcsin x.
     
  6. Jun 10, 2013 #5

    LCKurtz

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    So have you looked at the graphs of |arcsin(x)| and |arccos(x)|?
     
  7. Jun 11, 2013 #6

    utkarshakash

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    I get x= 0.707 by plotting the graph. But I need two values.
     
    Last edited: Jun 11, 2013
  8. Jun 11, 2013 #7
    Say, there is a point (x,y) where |asin(x)|=|acos(x)|=y, what does that say about sin(y) and cos(y)? Can you use that?
     
  9. Jun 11, 2013 #8

    LCKurtz

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    Can you use analysis to get the exact value?

    For a function continuous on a closed interval, where can the possible maximum and minimum points occur? What is the domain in this problem?
     
  10. Jun 12, 2013 #9

    utkarshakash

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    I am thinking it the other way. I can rewrite the original expression as
    [itex]\pi /2 \left( \pi ^2 /4 - 3sin^{-1} x cos^{-1} x \right) [/itex]
     
    Last edited: Jun 12, 2013
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