Let $f(x)=3 x^2+4 x g^{\prime}(1)+g^{\prime \prime}(2)$ and, $g(x)=2 x^2+3 x f^{\prime}(2)+f^{\prime \prime}(3)$ for all $x \in R$. Then,

all the above

We have,

$f(x)=3 x^2+4 x g^{\prime}(1)+g^{\prime \prime}(2)$         ......(i)

$g(x)=2 x^2+3 x f^{\prime}(2)+f^{\prime \prime}(3)$         ......(ii)

∴  $f^{\prime}(x)=6 x+4 g^{\prime}(1)$         ......(iii)

and, $g^{\prime}(x)=4 x+3 f^{\prime}(2)$         ......(iv)

$\Rightarrow f^{\prime \prime}(x)=6$         ......(v)

and, $g^{\prime \prime}(x)=4$         ......(vi)

Putting x = 1 in (iii) and (iv), we get

$f^{\prime}(1) =6+4 g^{\prime}(1)$ and $g^{\prime}(1)=4+3 f^{\prime}(2)$

$\Rightarrow f^{\prime}(1) =6+4\left\{4+3 f^{\prime}(2)\right\}$

$\Rightarrow f^{\prime}(1) =22+12 f^{\prime}(2)$

Putting x = 2 in (iv), we get

$g^{\prime}(2)=8+3 f^{\prime}(2)$

$\Rightarrow g^{\prime}(2)=8+3\left\{12+4 g^{\prime}(1)\right\}$       $\left[\begin{array}{l}\text { Putting } x=2 \text { in (iii) we get } \\ f^{\prime}(2)=12+4 g^{\prime}(1)\end{array}\right]$

$\Rightarrow g^{\prime}(2)=44+12 g^{\prime}(1)$

Putting x = 3 in (v) and x = 2 in (vi), we get

$f^{\prime \prime}(3)=6$ and $g^{\prime \prime}(2)=4$

$\Rightarrow f^{\prime \prime}(3)+g^{\prime \prime}(2)=6+4=10$