Maximum of $x+y+z$ subject to $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1$ via Lagrange multiplier

Question

$$\begin{array}{ll} \text{maximize} & x+y+z\\ \text{subject to} & \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1\end{array}$$

I get

$$1+a\lambda/x^2 = 0$$

$$1+b\lambda/y^2 = 0$$

$$1+c\lambda/z^2 = 0$$

But I don't get a value for $x$, $y$ and $z$. Can someone help me?

Answer 1

If we take $x=\frac{a}{1-2\varepsilon}, y=\frac{b}{\varepsilon}$ and $z=\frac{c}{\varepsilon}$ we can clearly see there is no maximum for $x+y+z$ by just letting $\varepsilon\to 0^+$. [I am assuming $a,b,c$ are positive] Before applying Lagrange multipliers, better to think if we are allowed to apply them, and if that is a wise choice.