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)

$5, 5$

Answer 2

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 E $(i)$ w.r.t. $'x'$ we get
$\frac{d^2P}{dx^2} = -2 < 0$
$\therefore P$ is maximum at $x = 5$
Hence, numbers are $5, 5$.