At the outset, it is known that x is a positive real number.
Determine the minimum value of:
x*floor(x) + floor(1/x) + x + 1/x + x*ceiling(x) + ceiling(1/x)
Well, Charlie beat me again, but here is my solution w/o looking at his.
Graphing the function leads to the observation that the domain of the minimum is a non-integer between 0 and 1, actually between 0.6 and 0.8. For that domain, the problem formula reduces to (term by term):<o:p></o:p>
0 + integer(1/x) + x + 1/x + x + integer(1/x) + 1 or<o:p></o:p>
2x + 2(integer(1/x)) + 1/x + 1<o:p></o:p>
For the noted domain of 0.6 to 0.8, integer(1/x) is always 1, so we want to minimize:<o:p></o:p>
F(x) = 2x + 1/x + 3<o:p></o:p>
Taking derivates and setting to zero: 2 + -(x^(-2)) = 0, or x = +- sqrt (1/2), of which +sqrt(1/2) is the value that minimizes the solution which is, with some algebra 2*sqrt(2) +3
Edited on April 17, 2023, 10:13 am
Edited on April 17, 2023, 10:14 am
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Posted by Kenny M
on 2023-04-17 10:09:40 |
Solution
