MathDB

Let

f

be a function defined on

(0, \infty )

as follows:

f(x)=x+\frac1x

Let

h

be a function defined for all

x \in (0,1)

h(x)=\frac{x^4}{(1-x)^6}

Suppose that

g(x)=f(h(x))

for all

x \in (0,1)

.
(a) Show that

h

is a strictly increasing function.
(b) Show that there exists a real number

x_0 \in (0,1)

such that

g

is strictly decreasing in the interval

(0,x_0]

and strictly increasing in the interval

[x_0,1)

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