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).