Ans : 5

$$\begin{gathered} \mathrm{f}\text{'}\left(\mathrm{x}\right) = \left\{5{\mathrm{x}}^4\sin \left(\frac{1}{\mathrm{x}}\right) - {\mathrm{x}}^3\cos \left(\frac{1}{\mathrm{x}}\right) + 10\mathrm{x}, \mathrm{x} < 0 \\ 0, \mathrm{x} = 0 \\ 5{\mathrm{x}}^4\cos \left(\frac{1}{\mathrm{x}}\right) + {\mathrm{x}}^3\sin \left(\frac{1}{\mathrm{x}}\right) + 2\mathrm{\lambda x}, \mathrm{x} > 0\right\} \\ \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right) =\left(20{\mathrm{x}}^3\sin \left(\frac{1}{\mathrm{x}}\right) - 5{\mathrm{x}}^2\cos \left(\frac{1}{\mathrm{x}}\right) - 3{\mathrm{x}}^2\cos \left(\frac{1}{\mathrm{x}}\right) - \mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right) + 10, \mathrm{x} < 0 \\ 0, \mathrm{x} = 0 \\ 20{\mathrm{x}}^3\cos \left(\frac{1}{\mathrm{x}}\right) + 5{\mathrm{x}}^2\sin \left(\frac{1}{\mathrm{x}}\right) + 3{\mathrm{x}}^2\sin \left(\frac{1}{\mathrm{x}}\right) - \mathrm{xcos}\left(\frac{1}{\mathrm{x}}\right) + 2\mathrm{\lambda }, \mathrm{x} > 0\right) \\ \mathrm{f}\text{'}\text{'}\left(0^{+}\right) = \mathrm{f}\text{'}\text{'}\left(0^{-}\right) \\ \Rightarrow 2\mathrm{\lambda } = 10 \\ \Rightarrow \mathrm{\lambda } = 5 \end{gathered}$$