Prove that the equation has at least a solution in the interval (0,1) 4x^3 + 3x^2 - 2x - 1 = 0

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll use the Rolle's theorem to prove that the given equation  has a root over the interval (0,1).

Let's see how:

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


According to the Rolle's rule,


 where c belongs to (0,1).

If f(x) is a Rolle function, then  f(1)=f(0).


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


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

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neela | High School Teacher | (Level 3) Valedictorian

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We  know that if f(x)  is a continuous function and f(a) and f(b) are of  different sign, then f(x) should cross the X axis for some value  c in the interval (a ,b).

So to have root for  f(x) = 4x^3+3x^2-2x-1  ,  f(0) and f(1) are of different signs.

f(0) = 4*0^3 +3*0^2-2*0 -1 =  -1 which is negative.

f(1) = 4*1^3+3*1^2-2*1 -1 = 4 ehich is positive.

So f(x) , being a continuous , should take all vaues between f(0) = -1 to f(1) = 4  for any x in the interval (0, 1).

Therefore there is one x = c  for which f(c) = 0. This implies f(x) has a root in (0 , 1)