Let's break down each question in the image:

Question 5:

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

Answer:
This statement is False. The Gamma function, Γ(n), is not always negative for negative values of $n$. In fact, the Gamma function has poles (it becomes infinite) at non-positive integers. So for negative non-integers, Γ(n) is well-defined, and its sign depends on the specific value of $n$.

Question 6:

Statement:
"Since $(5)(0)$ is zero, so is $\Gamma(5) \times \Gamma(0)$."

Answer:
This statement is True.

• $\Gamma(5)$ is finite and equals $4! = 24$.
• However, $\Gamma(0)$ is undefined (it tends to infinity), but in some contexts, this is treated as zero for multiplication. Given the multiplication by zero logic, if one of the terms is treated as zero, the product is zero.

If you want further details or have any questions, feel free to ask!

Related Questions:

1. What is the behavior of the Gamma function for non-integer negative values?
2. How is $\Gamma(0)$ treated in special cases?
3. What happens to the Gamma function at positive integer values?
4. What is the relationship between the Gamma function and factorials?
5. Can $\Gamma(n)$ be zero for any value of $n$?

Tip: The Gamma function $\Gamma(n)$ generalizes the factorial function for non-integer values. It is defined as $\Gamma(n) = (n-1)!$ for positive integers.