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Math Help - Optimazation Problems

  1. #1
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    Optimazation Problems

    This one is killing me:

    Find the point on the line y = 4x + 7 that is closest to the origin.

    I keep trying the distance formula and I end up with:
    d = [ 17x^2 + 56x + 49]^1/2
    because I substituted for y, getting it down to two variables like my teacher said. The solutions manual keeps saying I'm wrong because after I differentiate that I get a different answer. They do something weird where they square it and then differentiate to make it easier to work with. I get an ugly looking fraction when I differentiate my findings.
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  2. #2
    MHF Contributor Unknown008's Avatar
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    D = \sqrt{17x^2 +56x + 49}

    \dfrac{dD}{dx} = \dfrac12(17x^2 + 56x + 49)^{-\frac12}. (34x + 56)

    Set D' to 0.

    0 = (17x + 28)(17x^2 + 56x + 49)^{-\frac12}

    x = -\dfrac{28}{17}

    y = 4\left(-\dfrac{28}{17}\right) + 7 = \dfrac{7}{17}

    Are you getting this?
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