Question

If $a,b,c\in R^{+}$ such that $a+b+c=27$ , then the maximum value of $a^{2}b^{3}c^{4}$ is equal to

• A. $2^{8} \cdot 3^{10}$
• B. $2^{9} \cdot 3^{11}$
• C. $2^{10} \cdot 3^{12}$
• D. $2^{11} \cdot 3^{13}$

Answer 1

Answer: (C)

Consider nine numbers $\frac{a}{2},\frac{a}{2},\frac{b}{3},\frac{b}{3},\frac{b}{3},\frac{c}{4},\frac{c}{4},\frac{c}{4},\frac{c}{4}$
Using $AM\geq GM$ , we get,
$\frac{\frac{a}{2} + \frac{a}{2} + \frac{b}{3} + \frac{b}{3} + \frac{b}{3} + \frac{c}{4} + \frac{c}{4} + \frac{c}{4} + \frac{c}{4} }{9}\geq \left(\left(\frac{a}{2}\right)^{2} \left(\frac{b}{3}\right)^{3} \left(\frac{c}{4}\right)^{4}\right)^{\frac{1}{9}}$
$\Rightarrow \frac{a + b + c}{9}\geq \left(\frac{a^{2} b^{3} c^{4}}{2^{2} 3^{3} 4^{4}}\right)^{\frac{1}{9}}$
$\Rightarrow \left(\frac{27}{9}\right)^{9}\geq \frac{a^{2} b^{3} c^{4}}{2^{2} 3^{3} 2^{8}}$
$\Rightarrow a^{2} b^{3} c^{4} \leq 2^{10} \cdot 3^{12}$