I have a math homework where it's being asked to prove that :
$\forall a \geq 0,\sqrt{a}\leq\frac{1+a}{2}$
However, I don't have any idea how I should start this one...
Any idea ?
I have a math homework where it's being asked to prove that :
$\forall a \geq 0,\sqrt{a}\leq\frac{1+a}{2}$
However, I don't have any idea how I should start this one...
Any idea ?
Try expanding $ (\sqrt a - 1)^2 \geq 0 $
More generally, let $a,b\geq0$.
$(a-b)^2 \geq0$
$a^2-2ab+b^2 \geq0$
$a^2+2ab+b^2 \geq 4ab$
$(a+b)^2 \geq 4ab$
$\left(\frac{a+b}{2}\right)^2 \geq ab$
$\frac{a+b}{2} \geq \sqrt{ab}$