The non-negative values of b for which the function (16x³)/3 - 4bx² + x has neither maximum nor minimum in the range x > 0 is

Differentiating: f'(x) = 16x² - 8bx + 1. For no max or min in x > 0, f'(x) must not change sign for x > 0, which means f'(x) = 0 has no positive real roots. The discriminant is 64b² - 64 = 64(b² - 1). If b² - 1 < 0, i.e., |b| < 1, then f'(x) has no real roots and f'(x) > 0 always. For non-negative b, this gives 0 ≤ b < 1. At b = 0, f'(x) = 16x² + 1 > 0, no critical points. Hence 0 ≤ b < 1.