Question

Prove the inequalities. c+ 1/c ≥2 for c>0

Answer 1

$c+\dfrac1c=\dfrac{c^2+1}c=\dfrac{c^2-2c+1}c+2=\dfrac{(c-1)^2}c+2$

We have $(c-1)^2=0$ when $c=1$, and for all other values of $c$ we have $(c-1)^2>0$. So

$c+\dfrac1c=\dfrac{(c-1)^2}c+2\ge2$