Prove that there exists a unique real number x between 0 and

Prove that there exists a unique real number x between 0 and 1 such that x^3 + x - 1 = 0.

Let, f(x)=x^3+x-1

f is a continuous function for all real numbers

f(0)=-1

f(1)=1

Hence by Intermediate value theorem since f is continuous f must take all values in the interval (-1,1) for x in the interval: (0,1)

0 belongs to (-1,1). Hence for some x in (0,1)

f(x)=0

So we proved a root exists in the interval (0,1)

Now we need to prove it is unique

For this we apply Descartes sign rule

f(x) has only 1 sign change hence, at most 1 positive real root

f(-x) =-x^3-x-1 has no sign change so it has no negative real roots

We proved there is one positive real root in the interval (0,1) and hence it must be unique and the only real root for f(x)