SOLUTION: If an open box is made from a tin sheet 7 in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest b

We draw the little squares with sides x to cut out of each corner, 
which will divide each side of the 7"x7" square into parts x, 7-2x, 
and x inches as shown



We cut out the squares and draw in the base of the box (in green):



Now once those sides are bent upward the box will have the
dimensions: (7-2x) by (7-2x) by x.  So the volume, V,

   Volume = (length)·(width)·*height)

Let y = volume

    y = (7 - x)·(7 - x)·x

Multiply the first two factor out by FOIL:

    y = (49 - 28x + 4x²)·x

Then multiply the expreeion in parentheses by x:

    y = 49x - 28x² + 4x³
 
Let's draw the graph:



We want to find the value of x corresponding to that highest
peak point, which is where a tangent line is horizontal and
therefore has slope 0, as you see by the green horizontal
tangent line drawn below:



The derivative of a function is a formula for the slope of a tangent line 
drawn any any point (x,y) on the graph of that function.  We want to set 
that slope = 0.  Therefore we find the derivative and set it = 0 and solve 
for x: 

We find the derivative :

    = 49 - 56x + 12x²

We set that equal to 0:

   49 - 56x + 12x² = 0

Get it in descending order

   12x² - 56x + 49 = 0

That can be factored:

  (2x - 7)(6x - 7) = 0

Using the zero-factor property:

  2x - 7 = 0;  6x - 7 = 0
      2x = 7;      6x = 7
       x = ;       x = 
       x = 3.5;     x = 1.17 (rounded to two decimal places.

[Notice that the significance of the answer 3.5 is that if
we cut 3.5 inch squares out of the original square, there
will be nothing left!!! And so the volume is 0 then!  That
is the minimum volume, 0, when we cut all the tin away!  
Notice on the graph that the curve is tangent to the x-axis 
at x=3.6 and reaches a minimum value there.] 

So the maximum volume is found where x =  or 1.17 inches,
rounded to two decimal places.  
      
Edwin