Question

If the sum of the two numbers are fixed. Then, its multiplication will be maximum at which conditions?
(a) Each number is divisible by its sum.
(b) Each number is double of its sum.
(c) Each number is half of its sum
(d) Cannot be determined

Answer 1

Hint : The given problem revolves around the basic concepts of solving algebraic equations such as assuming the required parameters and then solving it by the given conditions. Then, by substituting the values obtained the required condition is obtained. Use $f'(x) = 0{\text{ }}$ to get the desired value.

Complete step-by-step answer :
Since, we have given that,
Any two numbers are fixed!
Therefore, let us assume that ‘ $a$ ’ and ‘ $b$ ’ be that two numbers respectively,
Hence, from the given condition that is sum of these two numbers are fixed
As a result, ‘ $S$ ’ be that sum of two numbers ‘ $a$ ’ and ‘ $b$ ’ respectively
We can write mathematically that,
$a + b = S$ ... (i)
That is $b = S - a$
Again, consider ‘ $b$ ’ as the function ‘ $a$ ’ that is ‘ $f(a)$ ’ respectively
Hence, the equation becomes,
$f(a) = ab$
Substituting $b = S - a$ , we get
$f(a) = a(S - a) \\\ \Rightarrow f(a) = aS - {a^2} \;$
Derive the above equation w.r.t. ‘ $a$ ’, we get
$f'(a) = S - 2a$
$\because$ We know that,
For any algebraic equation or expression, the maximum value can be represented if and only if $f'(x) = 0{\text{ where, }}x{\text{ is any variable}}$ ,
Hence, the equation becomes

$$\therefore f'(a) = S - 2a = 0 \\\ \Rightarrow S = 2a \;$$

Hence, we get
$\Rightarrow a = \dfrac{S}{2}$ … (ii)
As a result, equation (i) becomes,

\dfrac{S}{2} + b = S \\\ \Rightarrow b = S - \dfrac{S}{2} = \dfrac{S}{2} \; $$ … (iii) Form (ii) and (iii), it seems that Each number is half of its SUM i.e. $ a = b = \dfrac{S}{2} $ respectively! $ \Rightarrow \therefore $ The option (c) is correct! **So, the correct answer is “Option c”.** **Note** : It seems to that to find the maximum value in such analyze problems $ f'(x) = 0{\text{ where, }}x{\text{ is any variable}} $ is the condition to find it. One must be able to know basic mathematics to solve the problems in search of assumptions of the algebraic terms like in this problem, say, ‘ $ a $ ’ and ‘ $ b $ ’ so as to be sure of our final answer.