Does the identity $\sqrt{\frac{a}{2}}+\sqrt{\frac{a}{2}}\ge \sqrt{a}$ hold?

Question

Just out of curiosity, does the following identity hold?$$\sqrt{\frac{a}{2}}+\sqrt{\frac{a}{2}}\ge \sqrt{a},\quad a \gt 0$$

Thanks

Answer 1

$$\sqrt{\frac{a}{2}} + \sqrt{\frac{a}{2}} = 2\sqrt{\frac{a}{2}} = \frac{2}{\sqrt{2}}\sqrt{a}$$

Answer 2

Yes, you can distribute out $\sqrt a(\sqrt {\frac 12}+\sqrt {\frac 12})=\sqrt 2 \sqrt a \gt \sqrt a$

Answer 3

$$\sqrt{\frac{a}{2}} + \sqrt{\frac{a}{2}} = 2\sqrt{\frac{a}{2}} = \frac{2}{\sqrt{2}}\sqrt{a}\geq \sqrt{a}$$

Answer 4

$$\sqrt{\frac{a}{2}} + \sqrt{\frac a2} = 2 \frac{\sqrt a}{\sqrt 2} = \frac {2}{\sqrt 2} \sqrt{a} = \sqrt 2 \sqrt a = \sqrt{2a} > \sqrt a$$

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Added: perhaps your question originates from not remembering that $$(a + b)^2 \not\equiv a^2 + b^2\;?$$ If we do square each side of the proposed inequality (which we can do, without having to worry about whether doing so might change the direction of the inequality, since $a\gt 0$), then note: $$\left(\sqrt{\frac a2} + \sqrt{\frac a2}\right)^2\;\; \geq \; \;(\sqrt a)^2$$ $$ \iff \frac a2 + 2\cdot \frac a2 + \frac a2 \;\;\geq \;\; a $$ $$\iff 2a \geq a\quad \checkmark$$

Answer 5

Simple answer: yes. See this for explanation:

$$\sqrt{\frac{a}{2}}+\sqrt{\frac{a}{2}}\ge\sqrt{a} \; \wedge \; a>0$$ $$\Leftrightarrow \; 2\frac{\sqrt{a}}{\sqrt{2}}\ge\sqrt{a} \; \wedge \; a>0$$ $$\Leftrightarrow \; 2\frac{\sqrt{2}\sqrt{a}}{2}\ge\sqrt{a} \; \wedge \; a>0$$ $$\Leftrightarrow \; \sqrt{2}\sqrt{a}\ge\sqrt{a} \; \wedge \; a>0$$ $$\Leftrightarrow \; \sqrt{2}\ge1 \; \wedge \; a>0$$ Therefore, true.

Answer 6

Take $a=8b^2,$ where $b >0.$
Now $$\sqrt{\frac{a}{2}} + \sqrt{\frac{a}{2}} = 2\sqrt{\frac{a}{2}} = 2\sqrt\frac {8b^2}{2}=4b > 2\sqrt 2 b=\sqrt a.$$