prove that x54x1 has exactly one real solution in R be sure

prove that x^5+4x=1 has exactly one real solution in R. be sure to prove each claim.

f(x)=x^5+4x-1

f(0)=-1

f(1)=4>0

Since ,f is a continuous function so by intermediate value theorem there is a root of f in the interval

(0,1)

By Descartes sign rule there is only one positive real root for f as there is only one sign change

f(-x)=-x^5-4x-1

SInce f(-x) has no sign change so there are no negative real roots.

And we have found one real solution so that is the only real solution