UPSC NDA Math Model Paper 8

Let the pairs of positive numbers with sum 24 be: x and 24 - x.
Then let f(x) denotes the product of such pairs
i.e f(x)=x×(24−x)=24x−x2
Here we have to find that pair of number for which f(x) is maximum.
First we have to find f'(x)
⇒ f'(x) = 24 - 2x
Now let's find the roots of the equation f'(x) = 0
⇒ 2x = 24
⇒ x = 12
Now let's find out f"(x) i.e

d2(f(x))

dx2

⇒ f''(x) = - 2
Now evaluate the value of f''(x) at x = 12
⇒ f''(12) = - 2 < 0
As we know that according to second derivative test if f''(c) < 0 then x = c is a point of maxima
So, x = 12 is a point of maxima
So, when x = 12 then 24 - x = 12
Hence, the required numbers are 12 and 12