Minimum of Squares!

Algebra Level 3

If a+b=1anda,b>0 \color{#0C6AC7}a + \color{#456461}b = \color{magenta}{1} \quad and \quad \color{#0C6AC7}a , \color{#456461}b > \color{#624F41}0
Then minimum value\text{minimum value} of :

(a+1a)2+(b+1b)2 ? \left( \color{#0C6AC7}a \color{#EC7300}+ \dfrac{\color{magenta}1}{\color{#0C6AC7}a} \right)^{\color{#D61F06}2 }\color{#3D99F6}+ \left( \color{#456461}b \color{#EC7300}+ \dfrac{\color{magenta}1}{\color{#456461}b} \right)^{\color{#D61F06}2} \geq \ ?

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