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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives3 Marks
Question
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
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Answer

Let one number be x. Then, the other number is (16 - x).
Let S(x) be the sum of these number. Then,
S(x) = x3 + (16 - x)3 
$\Rightarrow$ S'(x) = 3x2 - 3(16 - x)2
$\Rightarrow$ S''(x) = 6x + 6(16 - x)
Now, S'(x) = 0
$\Rightarrow$ 3x2 - 3(16 - x)2 = 0
$\Rightarrow$ x2 - (16 - x)2 = 0
$\Rightarrow$ x2 - 256 - x2 + 32x = 0
$\Rightarrow$ x = 8
Now, S''(8) = 6(8) + 6(16 - 8)
= 48 + 48 = 96 > 0
Then, by second derivative test, x = 8 is the point of local minima of S.
Therefore, the sum of the cubes of the numbers is the minimum when the numbers are 8 and 16 - 8 = 8.

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