QJEEAdv09Mad04f: JEE Advanced 2009 Mathematics - Class 12 JEEAdv09Mad04f Solution: Let be a polynomial of degree having extremum at and . Then the …

Let us consider

$P(x) = a{x^4} + b{x^3} + c{x^2} + dx + e$

$P'(1) = P'(2) = 0$

$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^2} + P(x)} \over {{x^2}}}} \right) = 2$

$\Rightarrow P(0) = 0 \Rightarrow e = 0$

$\mathop {\lim }\limits_{x \to 0} \left( {{{2x + P'(x)} \over {2x}}} \right) = 2$

$\Rightarrow P'(0) = 0 \Rightarrow d = 0$

$\mathop {\lim }\limits_{x \to 0} \left( {{{2 + P''(x)} \over 2}} \right) = 2$

$\Rightarrow c = 1$

On solving, we get $a = 1/4,b = - 1$ . Thus,

$P(x) = {{{x^4}} \over 4} - {x^3} + {x^2} \Rightarrow P(2) = 0$