If a, b, c $> 0$ , and abc $= 5$ , then find the minimum value of $a^{3} + b^{3} + c^{3}$ .

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Solution

given $a b c = 5$
We know that A.M $\geq$ G.M for finite number of real terms
Consider the terms to be $a 3, b^{3}, c^{3}$
A.M $= \frac{a^{3} + b^{3} + c^{3}}{3}$
G.M $= \sqrt[3]{a^{3} b^{3} c^{3}} = a b c = 5$
A.M $\geq$ G.M
$a^{3} + b^{3} + c^{3} \geq 15$
The minimum value is $15$

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