Letting $\alpha_1,\dots,\alpha_{24} \in (0,\alpha]$ be the roots, we have $\prod_i \alpha_i=1$ and we want to maximize $\prod_i \vert 1-\alpha_i\vert$. It' easy to see that if we have $0<\alpha<1<\beta<\alpha$, then we increase the expression by increasing $\beta$ and decreasing $\alpha$, keeping the product constant. So we can assume that all variables that are bigger than $1$ are equal to $\alpha$. Moreover, it is easy to see that the expression is maximal when all of the variables that are smaller than $1$ are equal (otherwise again replace two of them by their geometric mean and just use AM-GM, only two factors change). So we are left with the situation of $24-k$ of the variables being equal to $\alpha$ and the remaining $k$ ones being equal to $\beta<1$ where $\beta^{k}\alpha^{24-k}=1$ and we need to prove that $\left(\frac{1-\beta}{\alpha-1}\right)^k \le \left(\frac{19}{5}\right)^5$. Let us first show that for fixed $k$, the expression is at most $\left(\frac{24-k}{k}\right)^k$. Indeed, we will show that if $t \in [0,1]$ and $\alpha^{1-t}\beta^t=1$, then $\frac{1-\beta}{\alpha-1} \le \frac{1-t}{t}$ (we can then choose $t=\frac{k}{24}$ to finish). But this is equivalent to $t\beta+(1-t)\alpha \ge 1$ which immediately follows from the condition by weighted AM-GM. Finally, we are only left to maximize $f(k)=\left(\frac{24-k}{k}\right)^k$ over all choices of $k \in \{1,2,\dots,23\}$. In general, the function $g(t)=\left(\frac{1-t}{t}\right)^t$ takes its maximum when $t \in (0,1)$ is the unique root of $\log \frac{1-t}{t}=\frac{1}{1-t}$, which happens to be $t_0 \approx 0.2178$, and $g$ is increasing for $t<t_0$ and decreasing for $t>t_0$. So it suffices to check that $f(4)<f(5)$ and $f(6)<f(5)$ i.e. that $\left(\frac{19}{5}\right)^5$ is bigger than both $5^4=625$ and $3^6=729$ which can be done by hand: It suffices to prove $\left(\frac{19}{15}\right)^5>3$. But the LHS is bigger than $\left(\frac{5}{4}\right)^5$ and the claim follows by noting that $5^5=3125>3072=3 \cdot 4^5$. In general, the maximum will occur when $k \approx t_0n$ where we replace $24$ by $n$.