Is it true that the product of two mixed numbers that are each between four and five is between 16 and 25

Yes! This is true. 

Here is the reasoning!
For something to be considered a mixed number it must have an integer and a proper fraction. 

An integer being the whole numbers, 4 and 5. 

A proper fraction is a fraction which is less than 1. So, 1/4 is a proper fraction. 

The maximum number we can have it 4 and some fraction that is infinitely close to 1. For simplicity sake we'll just use 9/10, although any fraction that is less than 1 would work. 

The reason the maximum is 4 is because if you had a fraction and an integer at 5 and a fraction then it would be beyond our limit of 4 and 5. 

4(9/10) 

We'll multiply two of these together, because this will show us our "maximum." 

4(9/10) * 4(9/10) = 49/10 * 49/10 = 2401/100 = 24.01 

As we get closer and closer to a fraction which equals 1, we see it get infinitely closer to 25. 

While this isn't something you'll be learning anytime soon (It's something you'll learn in either a pre-calculus class, or a calculus class. Either way you'll most likely learn about it either in an advanced math class in Highschool or College), but this is something called a limit. I won't go into details on it, but it will never reach 25, because we set it up so that the fraction can never equal 1. 

As for the minimum value. It would be an integer of 4 and a proper fraction that gets infinitely closer to 0. 

So, we'll use: 
4(1/100,000,000) = 400,000,001/100,000,000 = 4.00000001 * 4.00000001 = 16.00000008

Once again the number will never reach 16, but will get infinitely close to it.