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\noindent {\bf Problem statement} Consider line segments 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?

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\noindent {\bf How to get started} Suppose $C$ is a {\it positive}\/
number. What point on the curve has first coordinate equal to $C$?
What is the slope of the tangent line at that 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
{\it 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).

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