Prove that $\sqrt{a}\leq\frac{1+a}{2}$

Question

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 ?

Do you know the arithmetic mean- geometric mean inequality? If so, you can apply it directly to $1$ and $a$. — Dilip Sarwate, Sep 04 '12 at 19:59
I like the geometric proof as in http://upload.wikimedia.org/wikipedia/en/7/7c/SemicircleMeans.png — lhf, Sep 04 '12 at 21:34
Seems really nice though I can't understand why the line is $\sqrt{ab}$ long — Cydonia7, Sep 04 '12 at 21:37
@Skydreamer, all triangles with a vertex on the semicircle and basis the diameter are right triangles and the square of the height is the product of the two parts, which follows from Pythagoras's Theorem. — lhf, Sep 04 '12 at 22:32

score 16 · Accepted Answer · answered Sep 04 '12 at 19:58

16

Try expanding $$ (\sqrt a - 1)^2 \geq 0 $$

answered Sep 04 '12 at 19:58

AlbertH

Well, it was really simple... Sorry for the beginner's question. – Cydonia7 Sep 04 '12 at 20:01
That's OK - I have learned that, no matter how much I think I know about inequalities and how to prove them, what I don't know is far greater. – marty cohen Sep 04 '12 at 20:04
http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means – Iuli Sep 04 '12 at 21:15

score 7 · Answer 2 · answered Sep 04 '12 at 21:11

7

More generally, let $a,b\geq0$.

$$(a-b)^2 \geq0$$

$$a^2-2ab+b^2 \geq0$$

$$a^2+2ab+b^2 \geq 4ab$$

$$(a+b)^2 \geq 4ab$$

$$\left(\frac{a+b}{2}\right)^2 \geq ab$$

$$\frac{a+b}{2} \geq \sqrt{ab}$$

answered Sep 04 '12 at 21:11

Pedro

2 Answers2