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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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).

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