Question

Let -1 ≤ p ≤ 1, show that the equation 4x³ - 3x - p = 0 has a unique root in the interval [1/2, 1] and identify it.

Answer 1

f(x) has at least one real root between [1/2, 1]

f has only one real root between [1/2, 1].

To find a root, we observe f(x) contains 4x³ - 3x which is multiple angle formula for cos 3θ.