Math inequality proof If $a, b$ are positive real numbers and $a + b = 1$, prove that
$$
\left(a +\frac{1}{a}\right)^2 + \left(b +\frac{1}{b}\right)^2 \geq \frac{25}{2}
$$
Thank you.

Let $\displaystyle \phi(x) = \left(x + \frac{1}{x}\right)^2$. Since
$$\frac{d^2}{dx^2} \phi(x) = 2 + \frac{6}{x^4} > 0,$$
$\phi(x)$ is a strictly convex function for $x > 0$. 
By Jensen's inequality, we have
$$\left(a + \frac{1}{a}\right)^2 + \left(b + \frac{1}{b}\right)^2 = \phi(a)+\phi(b) \ge 2 \phi(\frac{a+b}{2}) = 2 \left(\frac12 + 2\right)^2 = \frac{25}{2}$$
