A Lagrange multiplier minimization problem,

Question

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,

Answer 1

$$F(x,y,z,\lambda)=x^{\frac{1}{p}}+y^{\frac{1}{q}}+z^{\frac{1}{r}}+\lambda (xyz-c)$$ 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.