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. 

Hint: Prove that $$\left(x+\frac{1}{x}\right)^2+\left(y+\frac{1}{y}\right)^2\geq \frac{25}{2}$$
Your inequality is equivalent to
$$(x^2+y^2)(1+\frac{1}{(xy)^2})\geq \frac{17}{2}$$
Since we have $$x^2+y^2=1-2xy$$
we get
$$1-2xy+\frac{1}{(xy)^2}-\frac{2}{xy}\geq \frac{17}{2}$$
Substituting $$xy=t$$ the inequality above is equivalent
$$\frac{(1-4t)(2+4t+t^2)}{2t^2}\geq 0$$
This is true,since we get by AM-GM $\frac{1}{2}\geq \sqrt{xy}$
