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Math Help - families of curves --- help!

  1. #1
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    families of curves --- help!

    heres the question:

    find a formula for a function of the form y = (ax^b)(lnx) where a and b are nonzero constants, which has a local maximum at the point (e^2,6e^-1)

    i tried to create 2 equations and set and solve for the variables, but its giving me a rediculous answer so i dont think its right.

    anyone know what to do?
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  2. #2
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    Sadly, you did not show your equations or your solution.

    You are right, though. The answers are not all that complicated.
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  3. #3
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    Quote Originally Posted by Paige05 View Post
    heres the question:

    find a formula for a function of the form y = (ax^b)(lnx) where a and b are nonzero constants, which has a local maximum at the point (e^2,6e^-1)

    i tried to create 2 equations and set and solve for the variables, but its giving me a rediculous answer so i dont think its right.

    anyone know what to do?
    y^{\prime}(x) = abx^{b - 1}ln(x) + ax^{b - 1}

    You wish that
    y^{\prime}(e^2) = ab(e^2)^{b - 1}ln(e^2) + a(e^2)^{b - 1} = 0

    And also
    6e^{-1} = a(e^2)^b \cdot ln(e^2) <-- From your original equation.

    These two equations simplify to
    2abe^{2(b - 1)} + ae^{2(b - 1)} = 0
    and
    \frac{6}{e} = 2ae^{2b}

    Can you solve these for a and b? (Note that you will also need to show that this is a local maximum, not a local minimum.)

    -Dan
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  4. #4
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    those are exactly the equations that i came up with, but when i solved for a and put that back into the second equation, i find it so hard to simplify and solve for b. im stuck.
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  5. #5
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    nevermind! i got it!

    thank you
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