Question

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

• A. $\frac{3^{10},2^4}{7^7}$
• B. $\frac{3^9,2^4}{7^7}$
• C. $\frac{3^9,2^5}{7^7}$
• D. $\frac{3^{10},2^5}{7^7}$

Answer 1

Answer: (A)

$\frac{3^{10},2^4}{7^7}$

we need $a^2{b}^3{c}^2$

So, taking $a+b+c=3$

$\frac{a}{2}+\frac{a}{2}+\frac{b}{3}+\frac{b}{3}+\frac{b}{3}+\frac{c}{2}+\frac{c}{2}=3$

Now use AM-GM

$\begin{array}{c}\frac{\frac{a}{2}+\frac{a}{2}+\frac{b}{3}+\frac{b}{3}+\frac{b}{3}+\frac{c}{2}+\frac{c}{2}}{7}\ge {\left(\frac{a^2}{2^2}\times \frac{b^3}{3^3}\times \frac{c^2}{2^2}\right)}^{1/7}\\ {\left(\frac{3}{7}\right)}^7\ge \frac{a^2{b}^3{c}^2}{2^2\times 3^3\times 2^2}\\ \frac{3^{10}\times 2^4}{7^7}\ge a^2{b}^3{c}^2\end{array}$