MathDB

Let

\alpha>1

and consider the function

f(x)=x^{\alpha}

for

x \ge 0

. For

t>0

, define

M(t)

as the largest area that a triangle with vertices

(0, 0), (s, f(s)), (t, f(t))

could reach, for

s \in (0,t)

. Let

A(t)

be the area of the region bounded by the segment with endpoints

(0, 0)

(t, f(t))

and the graph of

y =f(x)

. (a) Show that

A(t)/M(t)

does not depend on

t

. We denote this value by

c(\alpha)

. Find

c(\alpha)

. (b) Determine the range of values of

c(\alpha)

when

\alpha

varies in the interval

(1, +\infty)

.
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Problem Statement