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in Sets, relations and functions by (53.3k points)

Let f: R → R defined by f(x) = x3 + px2 + 3x + 2010. Then find the range of p for which f is a one-to-one function.

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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

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