Now just add the two areas together
Now, if we were to graph this, we would see that it has a point where it is minimized, meaning where the area is least. Because it is the least, the points around it must be greater than it, so this point is where the graph stops going down and begins going up. At this point, the tangent line is zero, meaning the slope is zero.
You know, then, that if you can find the equation of the slope, you can find out where it equals zero, and this will potentially be the point you are looking for.
We can find the equation of the slope by differentiating, so
Now, we want to find out what x value will cause A' to be equal to zero, so we set it equal to zero and solve for x.
Now, we have x = 8/7 as a point where our graph has a slope of zero, this means it might be the minimum, so we can check it in any of three ways. First is to plug it's value into our formula for area, then check a point on either side, and make sure it is less than those points, the second way is to check points on the derivative to make sure that it is decreasing to the point, and then increasing from the point (check a point before it should have a negative value, check a point after, it should have a positive value), and the third way is to take the derivative of the derivative (Area double prime) and plug the value into it's formula, if it is positive, then the derivative is increasing, so it will be a minimum, if it is negative, it will be a maximum.
I'll just plug it into our initial formula:
So we can see that the points around it are greater than it, which means it is a minimum value. Now we just need to check our endpoints, since they will not necessarily have tangents equal to zero, but could still be less than our value (not really in this problem, but it's good to get into the practice of checking, for other problems)
Obviously the lowest value x can be is zero, in which case the entire wire is used for the square. And the highest it can be is 2, as 2*8=16, and thus the entire wire is used for the rectangle. So we check these endpoints
So this means that these points are not less than x=8/7, so then x=8/7 must be the minimum value.