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Math Help - Critical numbers

  1. #1
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    Critical numbers

    For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it's decreasing.
    y= x square root (9-x^2)
    I did
    y = x sqrt(9-x^2)
    y' = (9-2x^2)/sqrt(9-x^2)
    0 = y' = 9-2x^2
    2x^2 = 9
    x^2 = 9/2
    x = sqrt(9/2) = 3/sqrt(2), -3/sqrt(2) (These are your critical numbers)
    Now just do a sign chart for y'
    -oo ---------- -3/sqrt(2) ------------ 3/sqrt(2) --------------- oo y'
    - - - - - - - - - - - - 0 + + + + + + + + 0 - - - - - - - - - - -
    The function is decreasing where it is - and increasing where it is +.
    So Increasing from (-3/sqrt(2),3sqrt(2)) and decreasing from (-oo,-3/sqrt(2)) and (3/sqrt(2),oo)
    A. sqrt(9/2) = 3/sqrt(2), -3/sqrt(2)
    B. (-3/sqrt(2),3sqrt(2))
    C. (-oo,-3/sqrt(2)) and (3/sqrt(2),oo)
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  2. #2
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    Your work appears correct and a graph confirms your answer. x*sqrt(9-x^2) - Wolfram|Alpha
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  3. #3
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    Thanks. Would it also be -3,3?
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  4. #4
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    Those should be the endpoints (instead of negative infinity and positive infinity). So, your intervals should be (-3,\frac{-3}{\sqrt{2}}) \cup (\frac{-3}{\sqrt{2}},\frac{3}{\sqrt{2}}) \cup (\frac{3}{\sqrt{2}},3).
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