Question

If $a,b\in R^{+}$ Find the minimum value of $E=\frac{16}{a^2}+\frac{b^2}{2}+\frac{a^2}{b}$

Answer 1

$\to E=\frac{8}{a^2}+\frac{8}{a^2}+\frac{b^2}{2}+\frac{a^2}{2}+\frac{a^2}{2b}$

Applying $A.M\ge G.M$

$\Rightarrow \frac{E}{5}\ge (\frac{8}{a^2}\times \frac{8}{a^2}\times \frac{b^2}{2}\times \frac{a^2}{2b}\times \frac{a^2}{2b}{)}^{1/5}$

$\Rightarrow E\ge 5(8{)}^{1/5}$

$\Rightarrow$ Minimum valume of E $=5(8{)}^{1/5}$