JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Application of Derivatives

Let

P (x)

be a real polynomial of degree 3 which vanishes at

x = - 3

. Let

P (x)

have local minima at

x = 1

, local maxima at

x = - 1

and

\int_{- 1}^{1} P (x) d x = 18

, then the sum of all the coefficients of the polynomial

P (x)

is equal to......

Answer: 8

Solution

A cubic with a local maximum at

x = - 1

and a local minimum at

x = 1

has

P^{'} (x) = k (x^{2} - 1)

with

k > 0

, so

P (x) = k (\frac{x ^{3}}{3} - x) + c

The condition

P (- 3) = 0

gives

- 6 k + c = 0

, hence

c = 6 k

. The odd part contributes nothing over

[- 1, 1]

, so

\int_{- 1}^{1} P (x) d x = \int_{- 1}^{1} 6 k d x = 12 k = 18 ⟹ k = \frac{3}{2}

Therefore

P (x) = \frac{x ^{3}}{2} - \frac{3 x}{2} + 9

, and the sum of all its coefficients is

P (1) = \frac{1}{2} - \frac{3}{2} + 9 = 8

The sum of the coefficients is

8

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Integer answer