The real number k for which the equation 2x³ + 3x + k = 0 has two distinct real roots in [0, 1] lies between 1 and 2.

If 2x³ + 3x + k = 0 has 2 distinct real roots in [0, 1], then f'(x) will change sign once in the interval. Let f(x) = 2x³ + 3x + k. Then f'(x) = 6x² + 3 > 0 for all x. Since f'(x) is always positive, f(x) is strictly increasing. A strictly increasing function can have at most one real root. Therefore, the equation 2x³ + 3x + k = 0 can have at most one real root in [0,1]. The statement that it has two distinct real roots in [0,1] is false. Thus, such a k does not exist.