OK, well the minimum volume is easy. If any of the values are zero, the volume will be zero. If "integer dimensions" is supposed to imply "positive non-zero integers", then your minimum volume is achieved by setting two sides to the minimum value of 1, and the remaining side to 48:
1 x 1 x 48 = 48cm^3
Now, there's a general rule that for any set of numbers with a fixed sum, their product is maximised if they're the same. For example, a(1-a) is max when a = (1-a) = 0.5. So the three integer values that are closest to each other, and sum to 50, are 17, 17 and 16:
17 x 17 x 16 = 4624cm^3
Which is the maximum volume for the solid (given integer dimensions).
If your question requires a proof of this, it likely involves some partial differentiation - if that's the case you can re-post under calculus specifying that - but I expect you're just required to find the logical answer.
(so of course the absolute difference is 4624 - 0 or 48 depending on whether 0 is allowed as a dimension; strictly speaking the question allows it since 0 is an integer, but I think it's more likely that they intended to mean positive integer and just worded it badly. So in that case the difference is 4576).
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