Let's break down both questions one by one.

Question 5:

"Γ(n) is always negative when n is negative."

The Gamma function, Γ(n), extends the factorial function to real and complex numbers. For positive integers, Γ(n) = (n-1)!. However, when $n$ is negative, the behavior of the Gamma function depends on whether $n$ is a negative integer or not:

• For negative integers, the Gamma function has singularities (it is undefined or diverges to infinity).
• For non-integer negative values, Γ(n) can be positive or negative.

So, the statement "Γ(n) is always negative when n is negative" is False, because the Gamma function is not well-defined for negative integers and can take positive values for some negative non-integers.

Question 6:

"Since $(5)(0) = 0$, so is $\Gamma(5) \times \Gamma(0)$."

This question is making a comparison using the Gamma function. Let's look at the facts:

• $\Gamma(5)$ is well-defined and equal to 4! (which equals 24).
• However, $\Gamma(0)$ is undefined (it tends toward infinity).

Since $\Gamma(0)$ is not a real number but is undefined, the product $\Gamma(5) \times \Gamma(0)$ does not follow the standard multiplication rules for numbers. Therefore, the conclusion that $\Gamma(5) \times \Gamma(0) = 0$ is False.

Do you want more details or have any questions?

Here are 5 related questions:

1. What is the general formula for the Gamma function?
2. Can the Gamma function take negative values for non-integer arguments?
3. Why is the Gamma function undefined at zero and negative integers?
4. How does the Gamma function extend the factorial concept?
5. What are some applications of the Gamma function in probability and statistics?

Tip: The Gamma function is used in various fields, especially in probability theory, as it generalizes the factorial concept to non-integers.