## 4 Results and Discussion  
The steady-state problem (9)-(12) is solved in a numerical manner by using the bvp4c programme from MATLAB, as described by [PERSON] et al. [48]. The bvp4c further requires an initial guess, which should reveal the attributes of the dual solutions and satisfy boundary conditions (12). Besides, the relative error tolerance on residuals has been set to \(10^{-5}\). A suitable finite value of \(\eta\to\infty\) had been opted, namely \(\eta=\eta_{\infty}=10\) for the upper and lower branch solutions. The numerical results are retrieved for varied values that involved dimensionless parameters, namely mixed convection parameter \(\lambda\), velocity slip parameter \(N\), temperature slip parameter \(K\), and magnetic parameter \(M\), in the presence of nanoparticles. On top of that, nanoparticles Cu and Ag have been considered with water as base fluids. In fact, in order to cover the limitation of experimental data, the choice of values is as determined by prior researchers. The value of Prandtl number is considered as Pr = 6.2 (water-based nanofluid), while the values of \(n\), \(\phi\), \(c\) and \(s\) are \(n=3\), \(\phi\) = 0.2, \(c=-1\) and \(s=1\), respectively. These values are fixed throughout this study (except for comparison purpose). Besides, the values of dimensionless parameters \(\lambda\), \(N\), \(K\), and \(M\) are given in the respective figures and tables. Next, in order to validate the present numerical method employed, the present results of this study are compared with findings obtained from prior works of [PERSON] et al. [35] and [PERSON] and [PERSON] [49]. Table 2 presents the comparative studies of the present results for reduced skin friction coefficient \(f^{\prime\prime}(0)\) for several values of \(M\) and \(n\), when \(c\) = 1 and \(\lambda=s=\phi=N=K=0\). Moreover, as noted in Table 2, it is interesting to observe that the present results are in very good agreement with the above mentioned works. Hence, one can claim that the present results are indeed correct and accurate. Nonetheless, interestingly, the present numerical results do indicate the existence of dual (upper and lower branch) solutions. As discussed earlier, the stability analysis is performed in this study in order to test the stability of the dual solutions. Therefore, the linear eigenvalue problem (31)-(34) is solved numerically using the MATLAB programme bvp4c. In addition, the minimum eigenvalues \(\gamma_{1}\), for several values of \(\lambda\), \(N\), \(K\), and \(M\), are given in Table 3. The results in Table 3 clearly show that \(\gamma_{1}>0\) is for upper branch solution, while \(\gamma_{1}<0\) reflects the lower branch solution. These results are consistent with those of other studies (see [4; 8; 9; 39; 47]) and suggest that the upper branch solution is indeed stable, but unstable for the lower branch solution. \begin{table}
\begin{tabular}{c c c c c} \hline \(M\) & \(n\) & **[PERSON] et al. [35]** & **[PERSON] and [PERSON] [49]** & **Present** \\ \hline
0 & 1 & \(-1.41421\) & \(-1.419135111\) & \(-1.414214\) \\  & 3 & \(-2.29719\) & \(-2.301331346\) & \(-2.297186\) \\
2 & 1 & \(-\) & \(-2.00000000\) & \(-2.000000\) \\  & 3 & \(-\) & \(-2.701221616\) & \(-2.701216\) \\ \hline \end{tabular}
\end{table} 
Table 2: Numerical values of \(f^{\prime\prime}(0)\) for several values of \(M\) and \(n\).