<s>[INST] computation of minimum value of a function  
How will we calculate the minimum value ilof the equation with the given condition.
 [/INST] Using Holder,s Inequality
$\displaystyle (x^2+8y^2+27z^2)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\geq \bigg[\bigg(x^2\cdot \frac{1}{x}\cdot \frac{1}{x}\bigg)^{\frac{1}{3}}+\bigg(8y^2\cdot \frac{1}{y}\cdot \frac{1}{y}\bigg)^{\frac{1}{3}}+\bigg(27z^2\cdot \frac{1}{z}\cdot \frac{1}{z}\bigg)^{\frac{1}{3}}\bigg]^{3}=(1+2+3)^3=216$
and equality hold when $\displaystyle x^3=8y^3=27z^3$
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