Question

Let $f\left(x\right)=x^2+x{g}^{\prime }\left(1\right)+g^{\prime \prime }\left(2\right)$ and $g\left(x\right)=x^2+x{f}^{\prime }\left(2\right)+f^{\prime \prime }\left(3\right)$. Then

• A. $f^{\prime }\left(1\right)=4+f^{\prime }\left(2\right)$
• B. $g^{\prime }\left(2\right)=8+g^{\prime }\left(1\right)$
• C. $g^{\prime \prime }\left(2\right)+f^{\prime \prime }\left(3\right)=4$
• D. all of these

Answer 1

The correct option is D all of these

$f\left(x\right)=x^2+x{g}^{\prime }\left(1\right)+g^{\prime \prime }\left(2\right)$
$g\left(x\right)=x^2+x{f}^{\prime }\left(2\right)+f^{\prime \prime }\left(3\right)$
Differentiating both the equations, we get
$f^{\prime }\left(x\right)=2x+g^{\prime }\left(1\right)$ and $g^{\prime }\left(x\right)=2x+f^{\prime }\left(2\right)$..............(1)
Putting $x=1$ in eqn (1), we get
$f^{\prime }\left(1\right)=2+g^{\prime }\left(1\right)$ and $g^{\prime }\left(1\right)=2+f^{\prime }\left(2\right)$
Then, $f^{\prime }\left(1\right)=4+f^{\prime }\left(2\right)$
Putting $x=2$ in eqn (1), we get
$f^{\prime }\left(2\right)=4+g^{\prime }\left(1\right)$ and $g^{\prime }\left(2\right)=4+f^{\prime }\left(2\right)$
$g^{\prime }\left(2\right)=4+4+g^{\prime }\left(1\right)$
Then, $g^{\prime }\left(2\right)=8+g^{\prime }\left(1\right)$
Differentiate eqn (1) w.r.t.x, we get,
$f^{\prime \prime }\left(x\right)=2$ and $g^{\prime \prime }\left(x\right)=2$ for all x
$f^{\prime \prime }\left(3\right)=2$ and $g^{\prime \prime }\left(2\right)=2$
$g^{\prime \prime }\left(2\right)+f^{\prime \prime }\left(3\right)=2+2=4$