It is proved that the equation 8x ^ 3-12x ^ 2 + 6x + 1 = 0 has at least one root in the interval (- 1,0)

It is proved that the equation 8x ^ 3-12x ^ 2 + 6x + 1 = 0 has at least one root in the interval (- 1,0)

Let f (x) = 8x ^ 3-12x ^ 2 + 6x + 1
f(-1)=-8-12-6+1=-25<0
f(0)=1>0
The function is continuous in the interval (- 1,0)
According to the mean value theorem, there is at least one point in the interval (- 1,0) such that 8x ^ 3-12x ^ 2 + 6x + 1 = 0
So the equation 8x ^ 3-12x ^ 2 + 6x + 1 = 0 has at least one root in the interval (- 1,0)