Prove that the equation 4x^3+3x^2=2x+1 has a solution 0<x<1

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To prove that the given equation has a root in the set (0,1), we'll apply Rolle's theorem.

We'll choose a Rolle function f:[0,1]->R,

f(x)=x^4+x^3-x^2-x

f(x)=x^4+x^3-x^2-x.

According to the Rolle's theorem:

f(1)-f(0)=f'(c)(1-0),

where c belongs to (0,1).

If f(1)=f(0) (condition valid only for a Rolle function)=> f'(c)=0

We'll differentiate the Rolle function and we'll get:

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

If f'(c)=0 ,then c is a root of f'(x), c belongs to (0,1). q.e.d

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