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Let p(x) be a polynomial of degree 4 hav...

Let `p(x)` be a polynomial of degree 4 having extremum at `x = 1,2` and `lim_(x->0)(1+(p(x))/x^2)=2.` Then find the value of `p(2).`

A

`-1`

B

0

C

1

D

2

Text Solution

Verified by Experts

The correct Answer is:
B
Let `p(x) = ax^(4) + bx^(3) + cx^(3) + dx + e `
`implies p.(x)=4ax^(3) + 3bx^(2) + 2cx + d `
`:. p.(1)=4a + 3b + 2c + d = 0` ........ (1)
and `p.(2) = 32a + 12 b + 4c + d =0` .......... (2)
Since , `lim_(x to 0)(1 + (p(x))/(x^(2)))=2` [given]
`:. lim_(x to 0) ( ax^(4) + bx^(3) + (c + 1)x^(2) + dx + e)/(x^(2))=2`
`implies c +1 =2 , d =0 , e=0 implies c=1`
From Equation (1) and (2) , we get
`4a + 3b = -2`
and 32a + 12b = -4
`implies a = (1)/(4) and b=-1 " " :. p(x) = (x^(4))/(4) - x^(3)+x^(2)`
`implies p(2) = (16)/(4) - 8 + 4 implies p(2) =0`
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