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

$0^2<0

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
$a ........................eqn(1)

Multiply both sides with a, we get,

$a^2 .....................eqn(2)

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

$ab ...................eqn (3)

from eqn(2) and eqn (3)

$a^2

4. Multiply both sides by $a$
$a since $a$ can be 0

Multiply both sides by $b$
$a

Hence $a^{2}\le ab

A counter example for $a^{2} is given by 11rdc11