Let \(\mathrm{f}(\mathrm{x})\) satisfy the requirement of lag ranger mean value theorm in \([0,2]\). If \(\mathrm{f}(0)=0\) and \(|\mathrm{f}(\mathrm{x})| \leq(1 / 2)\) for all \(\mathrm{x}\) in \(|0,2|\) then (a) \(\left|\mathrm{f}^{\prime}(\mathrm{x})\right| \leq 2\) (b) \(|\mathrm{f}(\mathrm{x})| \leq 1\) (c) \(\mathrm{f}(\mathrm{x})=2 \mathrm{x}\) (d) \(\mathrm{f}(\mathrm{x})=3\) for at least one \(\mathrm{x}\) in \((0,2)\)

Using the given information, we have that \(|\mathrm{f}(\mathrm{x})| \leq \frac{1}{2}\) on the interval \([0, 2]\). The LMVT states that there exists a value \(c\) such that \(\frac{\mathrm{f}(2) - \mathrm{f}(0)}{2-0} = \mathrm{f}^{\prime}(c)\). Given that \(\mathrm{f}(0) = 0\), we have: \(\left|\frac{\mathrm{f}(2)}{2}\right| = \left|\mathrm{f}^{\prime}(c)\right|\) is the maximum absolute value of the derivative on the interval. We already know that \(|\mathrm{f}(x)|\leq \frac{1}{2}\), so it follows that: $$\left|f'(c)\right| = \left|\frac{f(2)}{2}\right| \leq \frac{1}{2}$$ Since \(|\mathrm{f}^{\prime}(c)| \leq \frac{1}{2}\), we can conclude that option (a) is false.

(b) The statement claims that \(|\mathrm{f}(\mathrm{x})| \leq 1\) for all \(x\) in \([0,2]\), which is true as we already have \(|\mathrm{f}(\mathrm{x})| \leq \frac{1}{2}\), which is indeed less than or equal to \(1\). So, option (b) is true. (c) To prove or disprove this statement, we need to show whether \(\mathrm{f}(\mathrm{x})=2 \mathrm{x}\) on the interval \([0,2]\). However, we already know that the absolute value of the function is bounded by \(\frac{1}{2}\) on the interval \([0, 2]\). Thus, this function cannot satisfy the given condition, and option (c) is false. (d) To prove or disprove this statement, we must determine whether there exists at least one \(x\) in \((0,2)\) for which \(\mathrm{f}(\mathrm{x})=3\). Although we know that \(|\mathrm{f}(\mathrm{x})| \leq \frac{1}{2}\) for all \(x\) in this interval, we can directly show that \(|\mathrm{f}(x)|\leq 3\) cannot hold given the established bounds of the absolute value of the function. Therefore, option (d) is false. Various aspects have been briefly touched upon in the above steps, but based on our analysis we can preliminarily conclude as follows: (a) False (b) True (c) False (d) False