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).