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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^(2)+dx+e`
`rArr p'(x)=4ax^(3)+3bx^(2)+2cx+d`
`therefore p'(1)=4a+3b+2c+d=0" "` ...(1)
and `p'(2)=32a+12b+4c+d=0" "` ...(2)
Since, `lim_(x to 0) (1+(p(x))/(x^(2)))=2" "` [given]
`therefore lim_(x to 0) (ax^(4)+bx^(3)+(c+1)x^(2)+dx+e)/(x^(2))=2`
`rArr c+1=2, d=0, e=0 rArr c=1`
From Equations (1) and (2), we get
`4a+3b=-2`
and `32a+12b=-4`
`rArr a=(1)/(4) and b=-1 therefore p(x) =(x^(4))/(4)-x^(3)+x^(2)`
`rArr p(2)=(16)/(4)-8+4 rArr p(2)=0`
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