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so I have a function $f = 2\pi r h$ with $r, h$ as incognites. I want to minimize it.

The restriction

$g = \pi r^2 h-0.25$

The problema is that when I do the method I get an inconsistency like: $$ 2 = \lambda r $$ $$ 2= \lambda \pi r $$ Does someone know why this happen?

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HINT: Use $$\begin{cases} f_r =2\pi h= \lambda g_r=\lambda 2\pi rh\\ f_h =2\pi r= \lambda g_h=\lambda\pi r^2 \\ \pi r^2 h=0.25\end{cases}$$

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The underlying problem is inconsistent. You're trying to extremize the curved surface area of a cylinder while holding the volume fixed. But if you want the curved surface area large, you can make the top and bottom very small and then make the cylinder very tall; if you want the curved surface area small, you can make the top and bottom very large and then make the cylinder very short. In other words, by sending $r \to 0$ with $h=\frac{0.25}{\pi r^2}$, you obtain a cylinder of the appropriate volume with very large curved surface area. By sending $r \to \infty$ with $h=\frac{0.25}{\pi r^2}$, you obtain a cylinder of the appropriate volume with a very small curved surface area. And in fact the curved surface area is a strictly monotone function of $r$ once you set $h=\frac{0.25}{\pi r^2}$ so there can't be any local extrema, either.

This issue manifested in the inconsistency of the Lagrange conditions.

The more natural geometric problem includes the top and bottom of the cylinder in the objective function, and then has a critical point.

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