The absurdity of $\Gamma(x)$'s minimum, and can it be applied to the factorial?

Question

I know that the Gamma function can be used as a representation of the factorial, but, at the same time, it is an extrapolation of $x!$. The Gamma function is cool and all, but what are its applications to the factorial function as an extrapolation. Loosely speaking, if the factorial function gave $\Gamma(x)$ its popularity, what can $\Gamma(x)$ do back to the factorial?

Specifically, my question is: Does the minimum of $\Gamma(x)$ in $[1,2]$ have any implications on the "minimum" of $x!$ in $[0,1]$. Does $x!$ even have a defined minimum? I am looking for an intuitive answer, preferably able to be understood by someone with knowledge of a first-year calculus course.

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There are other questions on MathSE, but they almost all of them address the calculation of the minimum. Others talk about why the Gamma function has a minimum mathematically with proofs involving second derivatives for concavity, whereas I am looking for why it exists, intuitively, possibly with some nice geometric proofs or an elaboration on its applications.

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I searched the internet too, however, resources on implications of the Gamma function's minimum are minimal :). If there are no implications of $\Gamma(x)$ on $x!$, then any applications to real-world or other mathematical instances would be a good resource for a deeper understanding of the minimum.

Answer 1

There is no such thing as the "minimum of $x!$ on $[0, 1]$," because the factorial is only defined on the non-negative integers. We have $0! = 1! = 1$ and that's all.

Philosophically speaking, personally I regard the fact that the Gamma function takes on factorial values as sort of beside the point. People don't study the Gamma function because it extends the factorial, they study it because it appears a lot of different analytic situations (for example in the volume of an $n$-ball, or the functional equation for the zeta function). In some of those situations it happens to generalize an appearance of the factorial (for example the beta function; probably there's a simpler example here) but even when it doesn't it still shows up a lot.

Answer 2

There is no such thing as the "minimum of $x!$ on $[0, 1]$," because the factorial is only defined on the non-negative integers. We have $0! = 1! = 1$ and that's all.

Philosophically speaking, personally I regard the fact that the Gamma function takes on factorial values as sort of beside the point. People don't study the Gamma function because it extends the factorial, they study it because it appears a lot of different analytic situations (for example in the volume of an $n$-ball, or the functional equation for the zeta function). In some of those situations it happens to generalize an appearance of the factorial (for example the beta function; probably there's a simpler example here) but even when it doesn't it still shows up a lot.