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I am trying to minimize

$$x^p/p + y^q/q + z^r/r$$

subject to the constraint g=xyz=c, where c is a positive constant.

Also the function is defined for x,y,z non-negative only.

Then by the constraint equation we know that x,y,z must all be non-zero.

Solving

$$\nabla f = \lambda \nabla g$$

I currently have that

$$x^p = y^q = z^r = \lambda c$$

I am stuck at this point. I am tempted to say that this function has no minimum on xyz=c because by the string of equalities above there is no constraint on lambda. However that thinking must not be correct, because there is a part two to this question which uses the minimal value to prove an inequality.

Any hints or suggestions are welcome.

Thanks,

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$$F(x,y,z,\lambda)=x^{\frac{1}{p}}+y^{\frac{1}{q}}+z^{\frac{1}{r}}+\lambda (xyz-c)$$ Thus $$x=[(\frac{pqr}{pq+qr+pr}-1)\times c]^{1/p}$$ $$y=[(\frac{pqr}{pq+qr+pr}-1)\times c]^{1/q}$$ $$z=[(\frac{pqr}{pq+qr+pr}-1)\times c]^{1/r}$$

May be there are some errors in your deduction.

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  • $\begingroup$ Hi @user287754, thanks so much for your reply - I have now found my one critical point. But how will we know it is indeed a minimizer, without more critical points to compare it to? Thanks, $\endgroup$ – User001 Nov 8 '15 at 4:58
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    $\begingroup$ Calculate its Hesse matrix. If it is positive definite, then the point is a minimum value. $\endgroup$ – user287754 Nov 8 '15 at 5:04
  • $\begingroup$ Hi @user287754, thanks for the suggestion - I have a weird habit of thinking that concepts of unconstrained optimization do not apply to constrained optimization problems. So, I've computed the Hessian matrix easily enough and evaluated the matrix at the critical point. I get a diagonal matrix, so the eigenvalues of the Hessian can just be read off to determine whether the matrix is indeed positive-definite. $\endgroup$ – User001 Nov 9 '15 at 0:00
  • $\begingroup$ However, there is one slight difficulty: the diagonal entries are of the form $(p-1)c^{exponent}$, $(q-1)c^{exponent}$, and $(r-1)c^{exponent}$. We know that c is positive, by assumption. And, we also know that $p,q,r$ are fixed positive constants, by assumption. However, we wouldn't know what (p-1), (q-1), and (r-1) are. Subtracting one from each of $p,q,r$ could make some or all of them negative -- which would then lead to negative eigenvalues. What do you think? Thanks, @user287754 $\endgroup$ – User001 Nov 9 '15 at 0:00

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