Question

Let $f\left(x\right)=x^2+xg\text{'}\left(1\right)+g\text{'}\text{'}\left(2\right)$ and $g\left(x\right)=x^2+xf\text{'}\left(2\right)+f\text{'}\text{'}\left(3\right)$, then:

• A. $f\text{'}\left(1\right)=4+f\text{'}\left(2\right)$
• B. $g\text{'}\left(2\right)=g\text{'}\left(1\right)+4$
• C. $g\text{'}\text{'}\left(2\right)+f\text{'}\left(3\right)=8$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct answer

Given that: $f\left(x\right)=x^2+xg\text{'}\left(1\right)+g\text{'}\text{'}\left(2\right)$ and $g\left(x\right)=x^2+xf\text{'}\left(2\right)+f\text{'}\text{'}\left(3\right)$

$$\begin{gathered} f\left(x\right)=x^2+xg\text{'}\left(1\right)+g\text{'}\text{'}\left(2\right) \\ f\text{'}\left(x\right)=2x+g\text{'}\left(1\right)....\left(1\right) \end{gathered}$$

$$\begin{gathered} g\left(x\right)=x^2+xf\text{'}\left(2\right)+f\text{'}\text{'}\left(3\right) \\ g\text{'}\left(x\right)=2x+f\text{'}\left(2\right).....\left(2\right) \end{gathered}$$

Putting $x=1$ in equations (1) and (2):

$$\begin{gathered} f\text{'}\left(1\right)=2+g\text{'}\left(1\right) \\ g\text{'}\left(1\right)=2+f\text{'}\left(2\right) \end{gathered}$$

Putting $x=2$ in equation (1);

$$\begin{gathered} f\text{'}\left(2\right)=4+g\text{'}\left(1\right) \\ g\text{'}\left(2\right)=4+f\text{'}\left(2\right) \end{gathered}$$

Then,

$$\begin{gathered} g\text{'}\left(2\right)=4+4+g\text{'}\left(1\right) \\ \Rightarrow g\text{'}\left(2\right)=8+g\text{'}\left(1\right) \end{gathered}$$

Also,

$$\begin{gathered} f\text{'}\text{'}\left(x\right)=2,g\text{'}\text{'}\left(x\right)=2 \\ f\text{'}\text{'}\left(3\right)=2,g\text{'}\text{'}\left(3\right)=2 \\ \therefore g\text{'}\text{'}\left(x\right)+f\text{'}\text{'}\left(x\right)=2+2=4 \end{gathered}$$

Hence, the correct answer is Option (D).