Answer:
1)
, 2) The domain of S is
. The range of S is
, 3)
, 4)
, 5) 
Step-by-step explanation:
1) The function of the box is:




2) The maximum cutout is:




The domain of S is
. The range of S is 
3) The surface area when a 1'' x 1'' square is cut out is:


4) The size is found by solving the following second-order polynomial:




5) The equation of the box volume is:

![V = [w\cdot l -2\cdot (w+l)\cdot x + 4\cdot x^{2}]\cdot x](https://tex.z-dn.net/?f=V%20%3D%20%5Bw%5Ccdot%20l%20-2%5Ccdot%20%28w%2Bl%29%5Ccdot%20x%20%2B%204%5Ccdot%20x%5E%7B2%7D%5D%5Ccdot%20x)



The first derivative of the function is:

The critical points are determined by equalizing the derivative to zero:



The second derivative is found afterwards:

After evaluating each critical point, it follows that
is an absolute minimum and
is an absolute maximum. Hence, the value of the cutoff so that volume is maximized is:

The surface area of the box is:

