Find the maximum of $f(x,y,z) = xyz,$ under the following constraint: $$\frac1x +\frac1y + \frac1z = 1$$ $x,y,z > 0$
I have attempted this question by using Lagrange multipliers and have only found the solution $x=y=z=3.$ However, it is clear that that this is a minimum value as the solution $x=2, y=z=4$ gives a greater value of $\,xyz.$ How, then, do I go about finding the maximum value?