Question

If sum of two numbers is 24 and their product is as large as possible, then one of the number is

• A. 23
• B. 20
• C. 18
• D. None of these

Answer 1

Answer: (D)

Let one number be $x$. Then, the other number is $(24-x)$.
( $\because$ sum of the two numbers is 24$)$ Let $y$ denotes the product of the two numbers. Thus, we have
$y=x(24-x)=24x-x^2$
On differentiating twice w.r.t. $x$, we get
$\frac{dy}{dx}=24-2x\text{ and }\frac{d^{2}y}{d{x}^2}=-2$
Now, put $\frac{dy}{dx}=0$
$\Rightarrow 24-2x=0$
$\Rightarrow x=12$
At $x=12,\frac{d^{2}y}{d{x}^2}=-2<0$
By second derivative test, $x=12$ is the point of local maxima of $y$. Thus, the product of the numbers is maximum when the numbers are $12$ and $24-12=12$.
Hence, the numbers are $12$ and $12$.