Differential calculus to determine maxima and minima

is it 886.5526389cm^3

Yes. But your "reported" answer should be 890 cc (2 significant digits) or 887 cc (3 significant digits).

If this was a "practical" problem, the corner cut should have been 3.6 cm (36 mm). That would be the most cost efficient (practical) number. Then the volume would be:

V = 3.6* (30 - 7.2) * ( 18 - 7.2) = 886.464 = 886 cc
 
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is it 886.5526389cm^3

To answer your question, it's incorrect rounded out to the seventh decimal place, because
you used too much of a rounded value of x to start with.

\(\displaystyle 8 - \sqrt{19} \ \approx \ 3.64110105646 \ \ \) is an example (not necessarily the minimum number)
of digits that allows you to ensure you have it correctly rounded to seven decimal places.


The volume would be approximately 886.5526394 cubic centimeters if it were desired to go
out that far.


But look at the information in the post above about its significant digits.
 
You can check the answer you got by examining the properties of the first derivative. When the first derivative is zero, the function is either at a maximum or a minimum. Now, if we want a maximum, that means the function will be increasing up until that point, then decreasing afterwards. If the function is increasing over a given interval, what does that say about the first derivative over the same interval? And what does it mean when the function is decreasing? Use those properties and verify that the first derivative behaves as you'd expect it to around your x value of ~3.6
 
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