For real x , the function ( x - a ) ( x - b ) ( x - c ) will assume all real values provided

Step-by-step solution

Let, y = x 2 - a + b x + ab x - c ⇒ yx - cy = x 2 - a + b x + ab ⇒ x 2 - a + b + y x + a b + c y = 0 For real roots, D ≥ 0 ⇒ a + b + y 2 - 4 ab + cy ≥ 0 ⇒ a + b 2 + y 2 + 2 a + b y - 4 ab - 4 cy ≥ 0 ⇒ y 2 + 2 a + b - 2 c y + a - b 2 ≥ 0 which is true for all real values of y if D ≤ 0 ⇒ 4 a + b - 2 c 2 - 4 a - b 2 ≤ 0 ⇒ a + b - 2 c + a - b a + b - 2 c - a + b ≤ 0 ⇒ 2 a - 2 c 2 b - 2 c ≤ 0 ⇒ a - c b - c ≤ 0 ⇒ c - a c - b ≤ 0 ⇒ c must lie between a and b i.e., a ≤ c ≤ b or b ≤ c ≤ a