NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

Two non-negative numbers whose sum is $9$ and the product of the one number and square of the other number is maximum, are:

Choose the correct answer:

Explanation

Solution

Let the numbers be $A$ and $B$ .

Given: $A + B = 9 \Rightarrow A = 9 - B$

Let the function to maximize be $f = AB^{2}$ .

$f = (9 - B)B^{2} = 9B^{2} - B^{3}$

For maxima, the first derivative must be zero:

$f' = \frac{d}{dB}(9B^{2} - B^{3}) = 18B - 3B^{2} = 0$

$3B(6 - B) = 0$

So, $B = 0$ or $B = 6$ .

To check for maximum, we find the second derivative:

$f'' = \frac{d}{dB}(18B - 3B^{2}) = 18 - 6B$

At $B = 6$ :

$f'' = 18 - 6(6) = 18 - 36 = -18$

Since $f'' < 0$ at $B = 6$ , the function is maximum.

Finding $A$ :

$A = 9 - 6 = 3$

The numbers are $3$ and $6$ .

Correct Option: 2