How to get the $\mu_1 (x_1)$? I tried to use the derivative and equal it to zero or simply solve for u but I never get $\mu_1 (x_1) = 0$. I plotted the function and it does not seem to ever be equal to $0$ in this interval.
It seem like $\mu_1 (x_1)$ is at the lowest when $u=0$ but isn't itself equal to zero? I'm a bit lost. Might be missing something with the argument?
It is important to distinguish between the optimal value of a function and the optimal solution. In our case we are looking for the optimal solution $u^{\text{optimal}} := \mu_1(x_1)$.
Taking a look at the given optimization problem we want to solve to receive $\mu_1(x_1)$ we find that the objective function is continuous (w.r.t. $u$) and that we only consider $u \in [0,1]$ which is a compact set. Using the extreme value theorem (also called Weierstrass' Theorem) we know that minimum and maximum of this function exist within the interval $[0,1]$ which was already mentioned by @Gribouillis.
Looking at the plot you generated, it can easily be seen that for the interval $[0,1]$ the minimal function value is obtained at $u = 0$. Hence, we have $\mu_1(x_1) = 0$. Once again, keep in mind that this is the optimal solution and not the optimal value (which should be $4/6$).
Of course this result cannot only be obtained by taking a look at the plot. To receive this result from calculation, you have to check the boundary of the interval (so $u = 0$ and $u = 1$) and also need to check whether there are some points $u \in (0,1)$ where the first order necessary optimality condition holds. For all these points you would then have to check the function value to find out which of them is an optimal solution.
