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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Differentiation

JEE Advanced · Mathematics · Differentiation

Match the following List-I with List-II

Match the following List-I with List-II Match the following List-I with List-II

Options

  1. A4321\begin{array}{llll}4 & 3 & 2 & 1\end{array}
  2. B3421\begin{array}{llll}3 & 4 & 2 & 1\end{array}
  3. C3421\begin{array}{llll}3 & 4 & 2 & 1\end{array}
  4. D1234\begin{array}{llll}1 & 2 & 3 & 4\end{array}

Correct answer

C. 3421\begin{array}{llll}3 & 4 & 2 & 1\end{array}

Step-by-step solution

 (P) f′(x)=2x+g′(1),f′′(x)=2g′(x)=2x+f′(2),g′′(x)=2 at x=1,f′(1)=2+g′(1) and g′(1)=2+f′(2)⇒f′(1)=4+f′(2)\begin{aligned} & \text { (P) } \\ & f^{\prime}(x)=2 x+g^{\prime}(1), f^{\prime \prime}(x)=2 \\ & g^{\prime}(x)=2 x+f^{\prime}(2), g^{\prime \prime}(x)=2 \\ & \text { at } x=1, f^{\prime}(1)=2+g^{\prime}(1) \text { and } g^{\prime}(1)=2+f^{\prime}(2) \\ & \Rightarrow f^{\prime}(1)=4+f^{\prime}(2)\end{aligned}  (Q) ∴2f(x)f(y)=f(x−y)+f(x+y)\text { (Q) } \therefore 2 \mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})=\mathrm{f}(\mathrm{x}-\mathrm{y})+\mathrm{f}(\mathrm{x}+\mathrm{y}) Replacing x by y and y by x , then 2f(y)f(x)=f(y−x)+f(y+x)2 f(y) f(x)=f(y-x)+f(y+x) From equation (1) and (2), we get f(x−y)=f(y−x)f(x-y)=f(y-x) Put y=0y=0, then differentiate (R) f′(x)=3x2+2xf′(1)+f′′(2)⇒f′(1)′=3+2f′(1)+f′′(2)⇒f′(1)+f′′(2)=−3⇒ and f′′(x)=6x+2f′(1)∴f′′(2)=12+2f′(1)∴−2f′(1)+f′′(2)=12\begin{aligned} & f^{\prime}(x)=3 x^2+2 x f^{\prime}(1)+f^{\prime \prime}(2) \\ & \Rightarrow f^{\prime}(1)^{\prime}=3+2 f^{\prime}(1)+f^{\prime \prime}(2) \\ & \Rightarrow f^{\prime}(1)+f^{\prime \prime}(2)=-3 \\ & \Rightarrow \text { and } f^{\prime \prime}(x)=6 x+2 f^{\prime}(1) \\ & \therefore f^{\prime \prime}(2)=12+2 f^{\prime}(1) \\ & \therefore-2 f^{\prime}(1)+f^{\prime \prime}(2)=12 \end{aligned} (S) n(a+b)n−1=nan−1+nbn−1n(a+b)^{n-1}=n a^{n-1}+n b^{n-1}
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