How to prove that the equation 5x^4-4x^3-2x+1=0 has one root in the interval (0,1)?

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

To create
Rolle's theorem, we'll have to determine the anti-derivative of the function
5x^4-4x^3-2x+1.

Int (5x^4-4x^3-2x+1)dx = 5x^5/5 - 4x^4/4 -
2x^2/2 + x + C

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

f(x) = x^5 - x^4 - 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", that belongs to (0,1),
so that f'(c) = 0.

But f'(x) =
5x^4-4x^3-2x+1

Based on Rolle's theorem,
there is a root "c", of the equation 5x^4-4x^3-2x+1 = 0, in the interval
(0,1).