Question

If $a+b+c=3$ and $a > 0, b > 0, c >0$, then the greatest value of $a^{2} \,b^{3} \,c^{2}$ is

• A. $\frac{3^{10} \cdot 2^{4}}{7^{7}}$
• B. $\frac{3^{9} \cdot 2^{4}}{7^{7}}$
• C. $\frac{3^{8} \cdot 2^{4}}{7^{7}}$
• D. None of these

Answer 1

Answer: (A)

Taking A.M. and G.M. of $7$ numbers
$\frac{a}{2}, \frac{a}{2}, \frac{b}{3}, \frac{b}{3}, \frac{b}{3}, \frac{c}{2}, \frac{c}{2}$, we get
$\frac{2 \cdot \frac{a}{2}+3 \cdot \frac{b}{3}+2 \cdot \frac{c}{2}}{7} \geq\left[\left(\frac{a}{2}\right)^{2}\left(\frac{b}{3}\right)^{3}\left(\frac{c}{2}\right)^{2}\right]^{\frac{1}{7}}$
$\Rightarrow \frac{3}{7} \geq\left(\frac{a^{2} b^{3} c^{2}}{2^{2} 3^{3} 2^{2}}\right)^{\frac{1}{7}}$
$\Rightarrow \frac{3^{7}}{7^{7}} \geq \frac{a^{2} b^{3} c^{2}}{2^{2} \cdot 3^{3} \cdot 2^{2}} $
$\Rightarrow a^{2} \,b^{3} \,c^{2} \leq \frac{3^{10} \cdot 2^{4}}{7^{7}}$
$\therefore $ greatest value of $a^{2} \,b^{3}\, c^{2}=\frac{3^{10} \cdot 2^{4}}{7^{7}}$.