Constrained optimisation: Minimize $(x+\frac{1}{x})^2 +(y+\frac{1}{y})^2$ subject to the constrain $x+y=1$ > Minimize $(x+\frac{1}{x})^2 +(y+\frac{1}{y})^2$ subject to the constrain that $x+y=1$, where $x$ and $y$ are positive. I used Lagrange-optimization and proved that $x-\frac{1}{x^3} = y - \frac{1}{y^3}.$ But I don't seem to get the answer.

I didn't see a Lagrange multiplier approach in the linked answer so I think I should add this to the OP's answer:

The function $f(x) = x - {1 \over x^3}$ is strictly increasing as can be seen by taking its derivative and observing the result is positive. Strictly increasing functions are one-to-one, so if $f(x) = f(y)$ one has $x = y$. Since $x + y = 1$ here, this leads to $x = y = {1 \over 2}$, implying that $(x + {1 \over x})^2 + (y + {1 \over y})^2 \geq {25 \over 2}$.

Note that to apply Lagrange multipliers here you have to work on domains $x, y > \epsilon, x + y \leq 1$ for small $\epsilon > 0$ and then let $\epsilon \rightarrow 0$.