Let \(f^{\prime}(0)=-3\) and \(f^{\prime}(x) \leq 5\) for all real values of \(x\). The \(\mathrm{f}(2)\) can have possible maximum value as

Applying Lagrange's mean value theorem on interval \([0,2]\), we get there exist atleast one ' \(c\) ' \(\in(0,2)\) such that
\(\begin{array}{ll}
& \frac{\mathrm{f}(2)-\mathrm{f}(0)}{2-0}=\mathrm{f}^{\prime}(\mathrm{c}) \\
\therefore & \mathrm{f}(2)-\mathrm{f}(0)=2 \mathrm{f}^{\prime}(\mathrm{c}) \\
\therefore & \mathrm{f}(2)=\mathrm{f}(0)+2 \mathrm{f}^{\prime}(\mathrm{c}) \\
\therefore & \mathrm{f}(2)=-3+2 \mathrm{f}^{\prime}(\mathrm{c})
\end{array}\)
Given that \(\mathrm{f}^{\prime}(x) \leq 5\) for all \(x\)
\(\begin{array}{ll}
\therefore & \mathrm{f}(2) \leq-3+10 \\
\therefore & \mathrm{f}(2) \leq 7
\end{array}\)
\(\therefore \quad\) Largest possible value of \(\mathrm{f}(2)\) is 7 .