Prove that equation 6x^5-5x^4-2x+1=0 may only have one root in domain (0,1).

We'll use Rolle's function to prove that the given
equation has one root over the range (0,1).

To apply
Rolle's theorem for a Rolle's function, we'll have to determine the anti-derivative of
the function6x^5-5x^4-2x+1.

Int (6x^5-5x^4-2x+1)dx = 6x^6/6
- 5x^5/5 - 2x^2/2 + x + C

We'll simplify and we'll get the
Rolle's function:

f(x) = x^6 - x^5 - x^2 +
x

We'll calculate f(0) =
0

We'll calculate f(1) = 1-1-1+1 =
0

Since the values of the fuction, at the endpoints of
interval, are equal: f(0) = f(1) =>  there is a point "c", between the values 0
and 1, so that f'(c) = 0.

But f'(x) =
6x^5-5x^4-2x+1

Based on Rolle's theorem, there is a value c
that cancels out the equation.

That means
that there is one root "c", of the equation 6x^5-5x^4-2x+1 = 0, in the interval
(0,1).