I was trying to find the roots of the polynomial f(x) = 4x^3+3x^2-6x-1.
By Rolle's theorem, If there exists 2 different values of x, say x(1) and x(2) that returns a same value of f(x),
then at some point between x(1) and x(2), d/dx[f(x)] = 0. ---> (1)
Hence, I differentiate f(x) to get, d/dx[f(x)] = 12x^2+6x-6
By (1) 12x^2+6x-6 = 0
Solving for x, i get 2 roots, x = -1 and 1/2.
Now solve for f(x) = 4x^3+3x^2-6x-1 with x = 1 (by changing the sign) and surprisingly, f(1) = 0 !!,
seems to be one of the root of f(x).
Can someone help me to understand this ?

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