Question

Let f(x) be a polynomial of degree 4 having extreme values at x = 1 and x = 2. If `lim_(x rightarrow 0) ((f(x))/x^2 + 1)` = 3 then f(–1) is equal to ______.

Options

• `1/2`

• `3/2`

• `5/2`

• `9/2`

Answer 1

Let f(x) be a polynomial of degree 4 having extreme values at x = 1 and x = 2. If `lim_(x rightarrow 0) ((f(x))/x^2 + 1)` = 3 then f(–1) is equal to `underlinebb(9/2)`.

Explanation:

∵ f(x) has extremum values at x = 1 and x = 2

∵ f'(1) = 0 and f'(2) = 0

As, f(x) is a polynomial of degree 4.

Suppose f(x) = Ax⁴ + Bx³ + Cx² + Dx + E

∵ `lim_(x rightarrow 0)(f(x)/x^2 + 1)` = 3

⇒ `lim_(x rightarrow 0)(("A"x^4 + "B"x^3 + "C"x^2 + "D"x + "E")/x^2+1)` = 3 

⇒ `lim_(x rightarrow 0)("A"x^2 + "B"x + "C" + "D"/x + "E"/x^2 + 1)` = 3

As limit has finite value, so D = 0 and E = 0

Now A(0)² + B(0) + C + 0 + 0 + 1 = 3

⇒ c + 1 = 3

⇒ c = 2

f'(x) = 4Ax² + 3Bx² + 2Cx + D

f'(1) = 0

⇒ 4A(1) + 3B(1) + 2C(1) + D = 0

⇒ 4A + 3B = –4  ...(i)

f'(2) = 0

⇒ 4A(8) + 3B(4) + 2C(2) + D = 0

⇒ 8A + 3B = –2  ...(ii)

From equations (i) and (ii), we get

A = `1/2` and B = –2

So, f(x) = `x^4/2 - 2x^3 + 2x^2`

Therefore, f(–1) = `(-1)^4/2 - 2(-1)^3 + 2(-1)^2`

= `1/2 + 2 + 2`

= `9/2`

Hence f(–1) = `9/2`