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).