Let p x = x 2 + a x + b have two distinct real roots, where a , b are real number. Define g x = p x 3 for all real number x Then, which of the following statements are true? I. g has exactly two distinct real roots. II. g can have more than two distinct real roots. III. There exists a real number α such that g x ≥ α for all real x

Step-by-step solution

Let the given quadratic polynomial p x = x 2 + a x + b has two distinct real roots α and β , then p x = x 2 + a x + b = x - α x - β and since g x = p x 3 = x 3 - α x 3 - β let α = α 1 3 and β = β 1 3 then g x = x 3 - α 1 3 x 3 - β 1 3 = x - α 1 x - β 1 x 2 + α 1 x + α 1 2 x 2 + β 1 x + β 1 2 ∵ the discriminants of quadratic equations x 2 + α 1 x + α 1 2 and x 2 + β 1 x + β 1 2 are negative. ∴ &#