Question

For positive reals $a,b,c>0$, find the maximum value of $abc(a+b+c)$ using Cauchy-Schwarz inequality.

• A. $a^3+b^3+c^3$
• B. $2a^{3}b+2b^{3}c+2c^{3}a$
• C. $a^{2}b+b^{2}c+c^{2}a$
• D. None of these

Answer 1

The correct option is D None of these

as we know that $A.M.\ge GM$
$\Rightarrow \frac{a+b+c+(a+b+c)}{4}\ge {}^4\sqrt{a\ast b\ast c\ast (a+b+c)}$
$\Rightarrow (\frac{2(a+b+c)}{4}{)}^4\ge abc(a+b+c)$
$\Rightarrow (\frac{(a+b+c)}{2}{)}^4\ge abc(a+b+c)$
$\Rightarrow (\frac{(a+b+c{)}^4}{2^4})\ge abc(a+b+c)$
Therefore maximum value is $(\frac{(a+b+c{)}^4}{2^4})$