I'm pretty sure I sure the exact same problem on a physics forum, here's what I put:
Define
is also continous over since it's a polynomial.
Hence, by the intermediate value theorem .
Hence as required.
Okay, my homework is "Prove that there exists a positive real number x such that (x^3)=2."
and I have no clue how I can solve it. sigh.
Is there anyone who can me to solve it using least upper bound property??
Thank you !!
That's quite true, except I thought it was unnecessary since the question wants proof that the "number exists" not that the "number exists and is unique".
However, if i'm going to do something I might as well do it right!
Claim: is continuous over .
Let .
Let and define . When :
as required.
By algebra of continuity so is continuous on .
Claim: is strictly increasing.
is continuous on as proven above. Alternatively, is continuous on where and .
Subclaim: is differentiable on .
.
Hence a limit exists so the function is differentiable on .
is continuous on and differentiable on . By the Mean Value Theorem .
Since , and we get that as required.
If you let (b,c) be subsets of [1,2] then f(x) is injective over [1,2]. Hence if , has to be unique.