Question

If $x,y>0$then maximum value of product $xy(72-3x-4y)$ is

• A. $1155$
• B. $1152$
• C. $1122$
• D. $1123$

Answer 1

The correct option is B $1152$

$f(x,y)=xy(72-3x-4y)$

We know, $AM\ge GM$

Considering, $p=3x,q=4y,r=72-3x-4y$

So, $\frac{p+q+r}{3}\ge \sqrt{\left(\right(3x\left)\right(4y\left)\right(72-3x-4y){)}^{\frac{1}{3}}}$

$\Rightarrow \frac{p+q+r}{3}\ge \left(12xy\right(72-3x-4y){)}^{\frac{1}{3}}$

Cubing both sides,

$\frac{{24}^3}{12}\ge xy(72-3x-4y)$

$\Rightarrow 1152\ge xy(72-3x-4y)$