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JEE Advanced 2012Paper 1

Exam: JEE Advanced
Year: 2012
Paper: Paper 1
Subject: Mathematics
Chapter: Application of Derivatives

JEE Advanced 2012 · Paper 1 · Mathematics · Application of Derivatives

Let p(x) be a real polynomial of least degree which has a local...

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^{\prime}(0)

is

Correct answer

9

Step-by-step solution

Since,

p(x)

has a local maximum at

x=1

and a local minimum at

x=3

and

p(x)

is a real polynomial of least degree. Hence, let

p^{\prime}(x)=k(x-1)(x-3)=k

\left(x^{2}-4 x+3\right)\Rightarrow p(x)=k

\left(\frac{x^{3}}{3}-2 x^{2}+3 x\right)+c

Now,

p(1)=6

and

p(3)=2

\begin{array}{l} \Rightarrow \frac{4}{3} k+C=6 \text{\text{ and } } 0+C=2 \Rightarrow k=3 \\ \therefore \quad p^{\prime}(x)=3(x-1)(x-3) \Rightarrow p^{\prime}(0)=9 \end{array}

$

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