Suppose x is between consecutive positive integers n and n+1. Then [x]=n and we have
n(x^2+1)=x^3
or
x^3-n(x^2+1)=0
The LHS is a polynomial in x, at least inside the interval from n to n+1
Call P(x)=x^3-n(x^2+1) and check the endpoints
P(n)=n^3-n(n^2+1)= -n
this is always negative
P(n+1)=(n+1)^3-n((n+1)^2+1)=n^2+n+3
this is always positive
So by the intermediate value theorem there is at least solution somewhere between n and n+1
I didn't call this a full solution because I haven't shown there is only one solution in the interval.
|
|
Posted by Jer
on 2021-07-09 08:46:33 |
solution (except for one detail)
