How can I prove that $\frac{a}{2} + \frac{x}{a} \geq \sqrt{2x}$

Question

How can I prove the inequality

$$ \frac{a}{2} + \frac{x}{a} \geq \sqrt{2x} $$ for positive $a$ and $x$.

It seems simple, but I don't see a way to prove it.

Answer 1

$$\left(\sqrt{\frac{a}{2}}-\sqrt{\frac{x}{a}}\right)^2\geq0.$$

Answer 2

Since everyone here is positive, compare the squares: $$\Bigl(\frac{a}{2} + \frac{x}{a}\Bigr)^2=\frac{a^2}{4} + \frac{x^2}{a^2}+ x \geq 2x \iff \frac{a^2}{4} + \frac{x^2}{a^2}- x=\Bigl(\frac{a}{2}-\frac{x}{a}\Bigr)^2\ge 0. $$