Find the cubic polynomial in x which attains its maximum value 4 and m

1.	Find the cubic polynomial in x which attains its maximum value 4 and minimum value 0 at x = -1 and 1 respectively.
Answer» Let the cubic polynomial bey = f(x). Since it attains a maximum atx = -1 and a minimum at x = 1. dy/dx = 0 at x = -1 and 1 dy/dx = k(x + 1)(x + 1) = k(x² - 1) Separating the variables we have dy = k(x² – 1) dx ∫dy = k∫(x2 - 1) dx y = k((x³/3) - x) + c ... (1) When x = – 1, y = 4 and when x =1,7 = 0 Substituting the equation (1) we have 2k + 3c = 12; – 2k + 3c = 0 On solving we have k = 3 and c = 2. Substituting these values in (1) we get the required cubic polynomial y = x³ – 3x + 2.

Find the cubic polynomial in x which attains its maximum value 4 and minimum value 0 at x = -1 and 1 respectively.

Answer»

Let the cubic polynomial bey = f(x). Since it attains a maximum atx = -1 and a minimum at x = 1.

dy/dx = 0 at x = -1 and 1

dy/dx = k(x + 1)(x + 1) = k(x² - 1)

Separating the variables we have dy = k(x² – 1) dx

∫dy = k∫(x2 - 1) dx

y = k((x³/3) - x) + c ... (1)

When x = – 1, y = 4 and when x =1,7 = 0

Substituting the equation (1) we have

2k + 3c = 12; – 2k + 3c = 0

On solving we have k = 3 and c = 2.

Substituting these values in (1) we get the required cubic polynomial y = x³ – 3x + 2.