IIT-JEE 2012 · Paper 1 · Mathematics Q58

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MathematicsInteger+4Question 58 of 60

Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$ . If $p(1)=6$ and $p(3)=2$ , then $p'(0)$ is

Your answer

Answer and worked solution

Official answer: 9

Least degree. Two turning points need $p'$ of degree 2: $p'(x)=k(x-1)(x-3)$ , so
$p(x)=k\left(\frac{x^3}3-2x^2+3x\right)+c.$

Conditions. $p(3)=k(9-18+9)+c=c=2$ , and $p(1)=\frac43k+c=6$ , so $k=3$ . (Then $p''(1)=-2k<0$ : a maximum at 1, as required.)

Answer. $p'(0)=3k=9$ .

Answer: 9

Topics:Maxima and minima Polynomials and remainder theorem

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IIT-JEE 2012 · Paper 1, Mathematics question 58