Answer:
The absolute minimum of the surface area function on the interval is
The dimensions of the box with minimum surface area are: the base edge and the height
Step-by-step explanation:
We are given the surface area of a box where x is the length of the sides of the base.
Our goal is to find the absolute minimum of the the surface area function on the interval and the dimensions of the box with minimum surface area.
1. To find the absolute minimum you must find the derivative of the surface area () and find the critical points of the derivative (
).
Next,
There is a undefined solution and a real solution
. These point divide the number line into two intervals
and
Evaluate S'(x) at each interval to see if it's positive or negative on that interval.
An extremum point would be a point where f(x) is defined and f'(x) changes signs.
We can see from the table that f(x) decreases before , increases after it, and is defined at
. So f(x) has a relative minimum point at
.
To confirm that this is the point of an absolute minimum we need to find the second derivative of the surface area and show that is positive for .
and for we get:
Therefore S(x) has a minimum at which is:
2. To find the third dimension of the box with minimum surface area:
We know that the volume is 60 and the volume of a box with a square base is
, we solve for h
Substituting V = 60 and
The dimension are the base edge and the height