1. ## inequalities

0≤a<b, prove that a^2≤ab<b^2. Also, show by example that it does not follow that a^2<ab<b^2

2. For the example assume a = 0 so b>0

$\displaystyle 0^2<0<b^2$

which is false

3. Originally Posted by rmpatel5
0≤a<b, prove that a^2≤ab<b^2. Also, show by example that it does not follow that a^2<ab<b^2
$\displaystyle a<b$ ........................eqn(1)

Multiply both sides with a, we get,

$\displaystyle a^2<ab$ .....................eqn(2)

Now multiply both sides of eqn(1) with b, we get

$\displaystyle ab<b^2$ ...................eqn (3)

from eqn(2) and eqn (3)

$\displaystyle a^2<ab<b^2$

4. Multiply both sides by $\displaystyle a$
$\displaystyle a<b \Rightarrow a^{2}\le ab$ since $\displaystyle a$ can be 0

Multiply both sides by $\displaystyle b$
$\displaystyle a<b \Rightarrow ab<b^{2}$

Hence $\displaystyle a^{2}\le ab<b^{2}$

A counter example for $\displaystyle a^{2}<ab<b^{2}$ is given by 11rdc11