# Inequality Problem | Tomato subjective 83

This is a subjective problem from TOMATO based on inequality.

Problem: Inequality Problem

If ${\displaystyle{a}}$ and ${\displaystyle{b}}$ are positive real numbers such that, ${\displaystyle{a + b = 1}}$, prove that,
${\displaystyle{\left(a + {\frac{1}{a}}\right)^2 + \left(b + {\frac{1}{b}}\right)^2 {\ge} {\frac{25}{2}}}}$.

Solution: ${\displaystyle{\left(a + {\frac{1}{a}}\right)^2 + \left(b + {\frac{1}{b}}\right)^2 {\ge} {\frac{25}{2}}}}$
${\displaystyle{\Leftrightarrow}}$ ${\displaystyle{a^2 + b^2 + {\frac{1}{a^2}} + {\frac{1}{b^2}} {\ge} {\frac{17}{2}}}}$ ... (i)
Now ${\displaystyle{a + b = 1}}$ ${\displaystyle{\Rightarrow}}$ ${\displaystyle{a^2 + b^2 + 2ab = 1}}$
${\displaystyle{\Rightarrow}}$ ${\displaystyle{a^2 + b^2 {\ge} {\frac{1}{2}}}}$ ... (ii)
From (i) & (ii) we get to prove ${\displaystyle{{\frac{1}{a^2}} + {\frac{1}{b^2}} {\ge} {8}}}$
${\displaystyle{\Leftrightarrow}}$ ${\displaystyle{{\frac{a^2 + b^2}{a^2 b^2}} {\ge} {8}}}$
${\displaystyle{\Leftrightarrow}}$ ${\displaystyle{{\frac{1}{a^2 b^2}} {\ge} {4}}}$ [ as ${a^2 + b^2 {\ge} {\frac{1}{2}}}$ ]
${\displaystyle{\Leftrightarrow}}$ ${\displaystyle{1 {\ge} {4 a^2 b^2}}}$
${\displaystyle{\Leftrightarrow}}$ ${\displaystyle{1 {\ge} (a + b)^2}}$
Now this follows directly from the given condition ${\displaystyle{a + b = 1}}$.