Find max or min value of $ u=x^py^qz ^r $ subjected to the condition $a/x + b/y + c/z=1$ using Lagrange multipliers

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Let $(X,Y,Z) = (a/x, b/y, c,z)$. The problem is to extremize ($u$, a constant multiple of the reciprocal of) $U = X^p Y^q Z^r$ subject to $p \frac{X}{p}+q\frac{Y}{q}+r\frac{Z}{r}=1$. If $p,q,r$ are all positive, the weighted AM-GM inequality implies $U$ is maximized when $X/p = Y/q = Z/r$, so that $u$ is minimized at the corresponding values of $x,y,z$. If $p,q,r$ are all negative, the minimum of $u$ is obtained by the same argument.

$u$ can be made arbitrarily large when any of $p,q,r$ are positive, by taking extremely large values of the variable raised to that power in the formula defining $u$. And arbitrarily close to $0$ when any of $p,q,r$ are negative.

Conclusion. There exists a nontrivial one-sided bound exactly when all of $p,q,r$ have the same sign, the extremum unique and attained when $X,Y,Z$ are proportional to $p,q,r$. When $p, q, r$ are not all of the same sign, $u$ attains all values in $(0,+\infty)$, for positive $x,y,z$.

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Updated on November 08, 2020

Comments

  • rsk
    rsk almost 3 years

    1 Attaching what I have tried, found $x,y,z$ values. I'm struck on whether $u$ will be max or min.

    • rsk
      rsk about 7 years
      @amWhy hi, u Der. Can you plz help me this?
    • zyx
      zyx almost 7 years
      Use TeX (Mathjax) syntax for the question, not an image.