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Today's calculation of Integral 784

Source: 2012 Tokyo University of Science entrance exam/Architecture

February 9, 2012

calculusintegrationfunctionlogarithmsanalytic geometrygeometrylimit

Problem Statement

Define for positive integer

n

, a function

f_n(x)=\frac{\ln x}{x^n}\ (x>0).

In the coordinate plane, denote by

S_n

the area of the figure enclosed by

y=f_n(x)\ (x\leq t)

, the

x

-axis and the line

x=t

and denote by

T_n

the area of the rectagle with four vertices

(1,\ 0),\ (t,\ 0),\ (t,\ f_n(t))

and

(1,\ f_n(t))

.
(1) Find the local maximum

f_n(x)

.
(2) When

t

moves in the range of

t>1

, find the value of

t

for which

T_n(t)-S_n(t)

is maximized.
(3) Find

S_1(t)

and

S_n(t)\ (n\geq 2)

.
(4) For each

n\geq 2

, prove that there exists the only

t>1

such that

T_n(t)=S_n(t)

.
Note that you may use

\lim_{x\to\infty} \frac{\ln x}{x}=0.

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