What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve at some point?
E)None of these

Question

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve  at some point?
E)None of these

Quizplus · Accepted Answer

To find the shortest length of the line segment that is cut off by the first quadrant and is tangent to the curve, we need to find the equation of the tangent line at some point on the curve and find the point where this tangent line intersects the x-axis. Let the point of tangency be (x,y). Then, the equation of the tangent line is given by y-y(x) = y'(x)(x-x(x)), where y'(x) is the derivative of the curve at x.

The given curve is 11eaa3e5_55b9_cea5_9f8f_57182f425646_TB5971_11, and its derivative is 11eaa3e5_55b9_cea5_9f8f_57182f425646_TB5971_11. Setting y'(x) equal to its value at x = 1, we get y'(1) = 11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11.

At the point of tangency, the tangent line is also perpendicular to the radius vector from the origin to (x,y), which means that the slope of the tangent line is equal to -x/y. Setting this equal to y'(1), we get -x/y = 11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11, or y = -x/(11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11).

Now, we need to find the x-intercept of this line, which is given by setting y=0 in the equation above: 0 = -x/(11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11), or x=0. Therefore, the point of tangency is (0,0), and the length of the line segment cut off is simply the distance from (0,0) to the point of intersection of the curve and the x-axis.

To find this point, we set y=0 in the equation for the curve: 0 = 11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11x^3 + 11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11x, or x(11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11x^2 + 11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11) = 0. This gives us two possible x-values: x=0 and x=-11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11. Since the latter value is negative, it does not correspond to a point in the first quadrant. Therefore, the point of intersection is (11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11,0).

The length of the line segment cut off is simply the distance between (0,0) and (11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11,0), which is 11eaa3e5_55b9_f5b7_9f8f_a7d007c6638d_TB5971_11. Therefore, the answer is choice B.

What Is the Shortest Possible Length of the Line Segment