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. $'x'$ we get
$\frac{d^2P}{dx^2} = -2 < 0$
$\therefore P$ is maximum at $ x = 5$
Hence, numbers are $5, 5$.