Question

If we write $272$ as the sum of positive real numbers, so as to maximize their product, then $272$ is written as

• A. $272=2+2+\cdots \text{upto}136\text{terms}$
• B. $272=2.72+2.72+\cdots \text{upto}100\text{terms}$
• C. $272=34+34+\cdots \text{upto}8\text{terms}$
• D. $272=17+17+\cdots \text{upto}16\text{terms}$

Answer 1

The correct option is B $272=2.72+2.72+\cdots \text{upto}100\text{terms}$

To maximize the product, all of the numbers should be equal (using A.M-G.M inequality).
Let $272$ divided into $x$ equal parts. $(x\in N)$
So, we need to maximize
$y={\left(\frac{272}{x}\right)}^x$
Taking $\ln$ to both the sides,
$\Rightarrow \ln y=x(\ln 272-\ln x)$
Differentiate w.r.t $x$
$\Rightarrow \frac{1}{y}\cdot y^{\prime }=\ln 272-x\times \frac{1}{x}-\ln x$
To find critical point, $y^{\prime }=0$
$\Rightarrow \ln 272-1-\ln x=0$
$\Rightarrow x=\frac{272}{e}$
$\Rightarrow x=100[\because x\in N]$
Hence, the maximum product occurs when $272=2.72+2.72+\cdots \text{upto}100\text{terms}$