Question

If a + b + c = 18, find the maximum value of $a^3{b}^{2}c$ given that a, b & c are positive numbers.

• A. 78732
• B. 78736
• C. 78722
• D. 78746

Answer 1

The correct option is A 78732

Here we are not asked the maximum value of the product abc but the product $a^3{b}^{2}c$. To find the maximum value of $a^3{b}^{2}c$, we should know the value of a + a + a + b + b + c, but we don't. Therefore, we try to write the sum a + b + c in such a form that we can get the value of $a^3{b}^{2}c$. As a + b + c = 18
$\Rightarrow \frac{a}{3}+\frac{a}{3}+\frac{a}{3}+\frac{b}{2}+\frac{b}{2}+c=18$. These are 6 number whose product is $\frac{a^3{b}^{2}c}{108}$. The product will be maximum when $a^3{b}^{2}c$ will be maximum which will happen when all the 6 numbers are equal. Therefore, $\frac{a}{3}=\frac{b}{2}=c=\frac{18}{6}=3$
$\Rightarrow$ a = 9, b = 6, c = 3. So, the maximum value of $a^3{b}^{2}c$ is 78732.