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Question 883627: Find a positive integer for which the sum
of its recipricol and four times its square is the smallest
possible.
Found 2 solutions by stanbon, josgarithmetic:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Find a positive integer for which the sum
of its recipricol and four times its square is the smallest
possible.
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S = 1/x + 4x^2
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Take the derivative to get:
S' = -1/x^2 + 8x
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Solve:: -1/x^2 + 8x = 0
8x^3 = 1
x = 1/2
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Cheers,
Stan H.
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Answer by josgarithmetic(39620) (Show Source):
You can put this solution on YOUR website! Integer to find, n. Is this for a Calculus course, or College Algebra/Intermediate Algebra?
 and you want y as small as possible; and n must be an integer. Use derivative,  . This should be 0 at any extreme value for y, including any minimum.
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Finding extreme values for y
 ignoring the denominator because it has no importance for equating the derivative to zero.
Based on the difference of cubes formula,  .
This would have three possible solutions for n, but maybe only one of them would be our minimum n for y.
The binomial:
The quadratic:
Discriminant,  , not a real number.
Only that one, real, extreme value so must be our input for the minimum for y.
We want an INTEGER, so the nearest ones to  are 0 and 1.
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In fact for smallest POSITIVE value for y, choose n=1.
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