Consider the function v=g(x), where g(x) =x(6-2x)(8-2x), with x being the length of a cutout in cm and v being the volume of an open box in cm³, where x must between 0 and 3 cm. determine the maximum volume of the open box in cm³. round your answer to two decimal places. also, indicate or show what process you used to obtain this answer. (hint: graphing)

The maximum volume of the open box is 24.26 cm³

Step-by-step explanation:

The volume of the box is given as , where and .

Expand the function to obtain:

Differentiate  wrt  x to obtain:

To find the point where the maximum value occurs, we solve

Discard x=3.54 because it is not within the given domain.

Apply the second derivative test to confirm the maximum critical point.

,

This means the maximum volume occurs at .

Substitute into to get the maximum volume.

The maximum volume of the open box is 24.26 cm³

See attachment for graph.

.25

step-by-step explanation: