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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 to 0) [1+(p(x))/x^(2)]= 2`. Then, the value of p(2) is ………… .

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The correct Answer is:
`p(2) = 0`
Let `p(x) = ax^(4) + bx^(3) + cx^(2) + dx +e `
` rArr" " p'(x) = 4ax^(3) + 3bx^(2) + 2cx+ d`
`:. p'(1) = 4a+3b+2c+d= 0 ` ...(i)
` and p'(2) = 32a+ 12b + 4c+ d = 0` ...(ii)
Since, ` underset( x to 0) lim (1+(p(x))/x^(2))=2" "` [given]
`:. underset( x to 0) lim (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 Eqs. (i) and (ii), we get
4a+3b =- 2
and ` 32a+ 12b=- 4`
` rArr" " a=1/4 and b =-1`.
` :. " " p(x) = x^(4)/4 - x^(3) + x^(2) `
`rArr" p(2) = 16/4 - 8 +4`
`rArr" " p(2) = 0`
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