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