Final answer:
The shortest possible length that is cut off to the curve y =7/x at some point is infinite.
Step-by-step explanation:
Finding the point on the curve where the tangent line is perpendicular to the x-axis to determine the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y =7/x at some point.
This means that the slope of the tangent line is equal to zero.
To find this point, we take the derivative of the curve with respect to x and set it equal to zero.
Differentiating y = 7/x gives us (dy/dx) = -7/x².
Setting this equal to zero and solving for x gives us x = 0.
Therefore, the shortest possible length of the line segment is the distance between the x-axis and the point (x=0, y=7/0) on the curve.
However, since the curve has an asymptote at x=0, this point is not defined.
Therefore, the shortest possible length is infinite.