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)
Graphing the function shows the answer is around 0.7072.
The segment in question then is y = 1+x+1/x+x+2 = 3+2*x+1/x
Its derivative is 2-x^(-2).
Setting that to zero, 1/x^2 = 2
x = 1/sqrt(2)
The minimum value of the function is
3+sqrt(2)+sqrt(2)
= 3 + 2*sqrt(2)
Check via this program:
d=.0001
v=1/sqrt(2)
for x=v-d:d:v+d
3+2*x+1/x
end
finds
d =
0.0001
v =
0.707106781186547
ans =
5.82842715303446
ans =
5.82842712474619
ans =
5.82842715302646
The middle answer is the smallest.
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Posted by Charlie
on 2023-04-17 10:02:32 |
solution
