Explanation
\begin{aligned}
& \mathrm{Given:}f(x)=x^{3}+x^{2}f^{\prime}(1)+xf^{\prime\prime}(2)+f^{\prime\prime}(3) \\
& \Rightarrow f(x)=3x^{2}+2xf^{\prime}(1)+f^{\prime}(2) & & ...(\mathrm{A}) \\
& \Rightarrow f^{\prime}(1)=3+2f^{\prime}(1)+f^{\prime}(2) \\
& \Rightarrow f^{\prime}(1)+f^{\prime\prime}(2)=-3 & & ...(\mathbf{B}) \\
& \mathrm{Now},f(x)=6x+2f(1) \\
& \Rightarrow f^{\prime}(2)=12+2f^{\prime}(1) & & ...(\mathrm{C}) \\
& \therefore f^{\prime}(1)+12+2f^{\prime}(1)=-3 \\
& \Rightarrow3f^{\prime}(1)=-15\Rightarrow f^{\prime}(1)=-5 \\
& \Rightarrow f^{\prime\prime}(2)=12-10=2 \\
& \mathrm{Putting}x=10\mathrm{in}(\mathrm{A}), \\
& f(10)=3(100)+2(10)(-5)+2 \\
& =300-100+2=202
\end{aligned}
