Q77 / 160 · TS EAMCET 2003 · Mathematics

The sum of two numbers is 20 . If the product of the square of one number and cube of the other is maximum, then the numbers are :

  • Correct answer
    A
    12,8
  • B
    3,4
  • C
    9,12
  • D
    15,18

Solution

Let the numbers be $x, y$. $\therefore \quad x+y=20 \Rightarrow y=20-x$ Let $S=x^3(20-x)^2$ On differentiating w.r.t $x$, we get $\frac{d S}{d x}=2 x^3(20-x)(-1)+3 x^2(20-x)^2$ For maximum or minimum, put $\frac{d S}{d x}=0$ $\Rightarrow \quad x^2(20-x)(-2 x+60-3 x)=0$ $\Rightarrow \quad x=0, x=12$ or $x=20$ $x$ cannot be 0 or 20 When $x=12$, then $y=20-12=8$ The required numbers are 12,8 .

Hence, the correct answer is option: 'A'.

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