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Thread: Maximum value of a multivariable function

  1. #1
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    Maximum value of a multivariable function

    Problem:
    Find the maximum value of $\displaystyle \frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}}$ as $\displaystyle (x , y, z) $ are nonzero points in $\displaystyle \mathbb{R}^3$
    ========================

    I believe the maximum value of $\displaystyle \frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}}$ is the dot product of the vector a = (1, 2, 3) and the unit vector, b, in the direction of (x, y, z). So $\displaystyle \left\langle a, b \right\rangle = \sqrt{1^2+2^2+3^2} = \sqrt{14} $. Does this seem logical? Thank you for your time.
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  2. #2
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    Quote Originally Posted by Paperwings View Post
    Problem:
    Find the maximum value of $\displaystyle \frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}}$ as $\displaystyle (x , y, z) $ are nonzero points in $\displaystyle \mathbb{R}^3$
    let $\displaystyle u=<x,y,z>, \ v = <1,2,3>.$ then: $\displaystyle ||u||=\sqrt{x^2+y^2+z^2}, \ ||v||=\sqrt{14},$ and: $\displaystyle u \cdot v = x+2y+3z.$ now by Cauchy Schwartz: $\displaystyle |u \cdot v| \leq ||u|| \cdot ||v||.$

    thus: $\displaystyle |x+2y+3z| \leq \sqrt{14} \sqrt{x^2+y^2+z^2},$ which gives us: $\displaystyle -\sqrt{14} \leq \frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}} \leq \sqrt{14}.$ so the maximum and the minimum of your function are $\displaystyle \sqrt{14}$

    and $\displaystyle -\sqrt{14}$ respectively. Q.E.D.
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