Question

The sum of two numbers is $10$. Their product will be maximum when they are

• A. 3, 7
• B. 4, 6
• C. 5, 5
• D. 8, 2

Answer 1

Answer: (C)

Let one number be $x$ and second number be $(10-x)$.
According to the question,
$P=x(10-x)$, where $P$ is the product of numbers on differentiating both sides w.r.t. ‘$x$' we get
$\frac{dP}{dx}=x\frac{d}{dx}(10-x)+(10-x)\times \frac{d}{dx}(x)$
$=x\times (-1)+(10-x)\times 1$
$=-x-x+10$
$\frac{dP}{dx}=-2x+10...(i)$
For maximum or minimum value,
$\frac{dP}{dx}=0$
$\Rightarrow -2x+10=0$
$\Rightarrow x=5$
Now, on differentiating Eq. $(i)$ w.r.t. ${}^{\prime }{x}^{\prime }$ we get
$\frac{d^{2}P}{d{x}^2}=-2<0$
$\therefore P$ is maximum at $x=5$
Hence, numbers are $5,5$.