We have
f(x) = x3 + px2 + 3x + 2010
⇒ f'(x) = 3x2 + 2px + 3
For f(x) to be one-to-one function, we need to have f′(x) ≥ 0 or ≤ 0. Here, f ′(x) is the quadratic expression and coefficient of x2 > 0 so that f′(x) ≥ 0.
D ≤ 0
⇒ 4p2 - 36 ≤ 0
⇒ p2 ≤ 9 ⇒ -3 ≤ p ≤ 3