Dimostrare che $\dfrac{a}{2}+\dfrac{2}{a}\ge 2$ per $a>0$

Text

Prove that, for every real number $a>0$, $\frac{a}{2}+\frac{2}{a}\ge 2,$ and determine when equality holds.

Solution

Reduce the difference to a single fraction: $\frac{a}{2}+\frac{2}{a}-2=\frac{a^2+4-4a}{2a}=\frac{(a-2)^2}{2a}.$ Since $a>0$, the denominator is positive and the numerator $(a-2)^2\ge 0$; hence the fraction is $\ge 0$, i.e. $\dfrac{a}{2}+\dfrac{2}{a}\ge 2$. Equality holds if and only if $(a-2)^2=0$, i.e. $a=2$. $\boxed{\frac{a}{2}+\frac{2}{a}\ge 2,\qquad =\ \text{iff } a=2.}$