Question

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.

विकल्प

• 12

• –12

• –24

• 6

Answer 1

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to –12.

Explanation:

Let P(x) = ax³ + bx² + cx + d

P'(x) = 3ax² + 2bx + c

Max at x = 1 and Min at x = 2

⇒ P'(x) = 0 at x = 1, x = 2

3a + 2b + c = 0  ...(i)

12a + 4b + c = 0  ...(ii)

(i) – (ii)

⇒ a = `-2/9b`  ...(A)

Max of P(x) is 8 at x = 1

8 = a + b + c + d  ...(iii)

Min 4 at x = 2

4 = 8a + 4b + 2c + d  ...(iv)

Equation (iii) – (iv),

4 = –7a – 3b – c

From (A), 

4 = `14/9b - 3b - c`

4 = `(-13)/9 b - c`

c = `-4 - 13/9b`  ...(v)

From (i), using A, equation (v)

3a + 2b + c = 0

`3(-2/9b) + 2b - 4 - 13/9b` = 0

`(-2)/3b + 2b - 13/9b` = 4

`(-b)/9` = 4

b = –36

a = `(-2)/9b = (-2)/9 xx (-36)` = 8

c = `-4 - 13/9(-36)`

c = 48

a + b + c + d = 8

8 – 36 + 48 + d = 8

⇒ d = –12

P(x) = ax³ + bx² + cx + d = 8

⇒ P(x) = 8x³ – 36x² + 48x – 12

⇒ P(0) = –12