consider line segment which are tangent to a point on the right half(x>0) of the

Question

consider line segment which are tangent to a point on the right half(x>0) of the curve y=x^2+1 and connect the tangent point to the x-axis. several are displayed in the diagram to the right. if the tangent point is close to the y-axis the line segment is long. if the tangent point is far from the y-axis the line segment is also very long. which tangent point has the shortest line segment?

How to get started:suppose C is a positive number. what point on the curve has first coordinate equal to C? what is the slope of the tangent line at the point? find the x-intercept, of the resulting line. compute the distance between the point on the curve and the x-intercept, and find the minimum of the square of that distance(minimizing the square of a positive quantity gets the same answer as minimizing the quantity, and here we get rid of a square root).

Explanation / Answer

1)If the point (a, a^2+ 1) is on the parabola, the tangent at this point has slope y'(a) = 2a, so the tangent line has equation

y - (a

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