Practice question on Correspondences and maximization
I think the easiest way is to notice that since the problem is increasing in both arguments, we can assume $px_1 + x_2 = 1$ the budget binds (at least for $p \not =0$).
Substituting our constraint into the objective function, we have:
$$
\max_{x_1} x_1 + 5(1 - p x_1) = x_1(1 - 5p) + 5
$$
If $1 - 5p&lt;0$, we choose the smallest possible $x_1$, so $x_1 = 0$ and thus $x_2 = 1$.
If $1 - 5p = 0$, any $x_1$ would maximise the above.
If $1 - 5 p &gt; 0$, you choose the biggest $x_1$ possible, so $x_2 = 0$ and $x_1 = \frac{1}{p}$.
If $p = 0$, there is no constraint on $x_1$, and your objective function is unbounded in $x_1$, so you would choose $x_1 = \infty$, so no solution exists.