# inequalities

• Sep 10th 2008, 06:35 PM
rmpatel5
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
• Sep 10th 2008, 06:42 PM
11rdc11
For the example assume a = 0 so b>0

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

which is false
• Sep 10th 2008, 06:58 PM
Shyam
Quote:

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\$
• Sep 10th 2008, 07:57 PM
acc100jt
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