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On the coordinate plane, Let

C

be the graph of

y=(\ln x)^2\ (x>0)

and for

\alpha >0

, denote

L(\alpha)

be the tangent line of

C

at the point

(\alpha ,\ (\ln \alpha)^2).

(1) Draw the graph.
(2) Let

n(\alpha)

be the number of the intersection points of

C

and

L(\alpha)

. Find

n(\alpha)

.
(3) For

0<\alpha <1

, let

S(\alpha)

be the area of the region bounded by

C,\ L(\alpha)

and the

x

-axis. Find

S(\alpha)

.
2010 Tokyo Institute of Technology entrance exam, Second Exam.

Problem Statement