Question

A polynomial f(x) of degree 4 with leading co-efficient unity and also has a local max or min at x = 1, 2, 3 respectively. If f(0) = 2, find the polynomial f(x).

Answer 1

Let f'(x) = (x – 1)(x – 2)(x – 3)

⇒ f'(x) = x³ – 6x² + 11x – 6

⇒ f(x) = x⁴/4 – 6(x³/3)  + 11(x²/2)  – 6x + b

⇒  f(x) = x⁴/4 – 2x³ +  11/2x² – 6x + b

f(0) = 2 gives b = 2

Hence, the polynomial function is

f(x) = x⁴/4 – 2x³ +  (11/2)x² – 6x + 2

⇒  f(x) = 1/4(x⁴ – 8x³ + 44x² – 24x + 8)