Question

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)dx` = 18, then the sum of all the coefficients of the polynomial P(x) is equal to ______.

Options

• 6

• 7

• 8

• 9

Answer 1

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)dx` = 18, then the sum of all the coefficients of the polynomial P(x) is equal to 8.

Explanation:

P'(x) is a second-degree polynomial and P(x) has local minima and local maxima at 1 and –1 respectively.

Let P'(x) = a(x – 1)(x + 1) = ax² – a

On integrating, we get

`\implies` P(x) = `a/3. x^3 - ax + b`

Now `int_-1^1 p(x) dx` = 18

`\implies int_-1^1(a/3. x^3 - ax + b)dx` = 18

`\implies` 2b = 18

`\implies` b = 9

Since P(x) is vanish at x = –3

∴ P(–3) = 0

`\implies a/3(-3)^3 - a(-3) + 9` = 0

`\implies` a = `3/2`

So, P(x) = `1/2x^3 - 3/2x + 9`

`\implies` Sum of all the coefficients

= `1/2 - 3/2 + 9`

= 8