Question

Find the cubic polynomial P(x) such that P(x) has maximum at x = –1 and P'(x) has minimum at x = 1. P(–1) = 10, P(1) = – 6. Also find the distance between the points of local maximum and local minimum of the curve.

Answer 1

Let the polynomial be

f(x) = ax³ + bx² + cx + d

According to the given conditions,

f(–1) = – a + b + c + d = 10

f(1) = a + b + c + d = 6

Also, f'(–1) = 3a – 2b + c = 0

⇒ f''(1) = 6a + 2b = 0

⇒ a + 3b = 0

On solving, we get,

a = 1, b = – 3, c = – 9, d = 5

Thus, f(x) = x³ – 3x² – 9x + 5

⇒ f'(x) = 3x² – 6x + 9

⇒ f'(x) = 3(x² – 2x – 3) = 3 (x + 1)(x – 3)

Therefore, x = –1 is point of maxima and x = 3 is the point of minima

Now, f(–1) = –1 – 3 + 9 + 5 = 10, pt is (–1, 10)

f(3) = 27 – 27 – 27 + 5 = –22, pt is (3, –22)