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Home  >>  CBSE XII  >>  Math  >>  Application of Derivatives
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Find two positive numbers x and y such that their sum is 16 and sum of whose cubes is minimum.

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1 Answer

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Toolbox:
  • For greatest area, maximise area.
  • Working rule for maxima minima.
  • Find the function to be maximised or minimised
  • Convert the function into single variable
  • Apply the conditions for maximum or minimum i.e., first derivative = 0 ( w.r.to the variable converted) and second derivative positive for minima and negative for maxima.
  • Answer the question.
Step 1
Given $ x+y=16$
and $ s=x^3+y^3$ is minimum.
$ s = x^3+(16-x)^3$
Step 2
$ \large\frac{ds}{dx}=3x^2-3(16-x)^2=0$
$ \Rightarrow x = 8\: \: y = 8$
Step 3
$ \large\frac{d^2s}{dx^2}=6x+6(16-x)$ is positive at $ x = 8$
$ \Rightarrow s$ is minimum.
Ans : $ x = 8, \: \: y = 8$
answered Apr 12, 2013 by thanvigandhi_1
 

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