MathDB

The functions

f,g:[0,1]\to\mathbb R

are given with the formulas

f(x)=\sqrt[4]{1-x},\enspace g(x)=f(f(x)),

and

c

denotes any solution of

x=f(x)

.
(a) i. Analyze the function

f(x)

and draw its graph. Prove that the equation

f(x)=x

has the unique root

c

satisfying

c\in[0.72,0.73]

. ii. Analyze the function

f'(x)

. Let

M_1

and

M_2

be the points of the graph of

f(x)

with different

x

coordinates. What is the position of the arc

M_1M_2

of the graph with respect to the segment

M_1M_2

? iii. Analyze the function

g(x)

and draw its graph. What is the position of that graph with respect to the line

y=x

? Find the tangents to the graph at points with

x

coordinates

0

and

1

. iv. Prove that every sequence

\{a_n\}

with the conditions

a_1\in(0,1)

and

a_{n+1}=f(a_n)

for

n\in\mathbb N

converges. [hide=Official Hint]Consider the sequences

\{a_{2n-1}\},\{a_{2n}\}~(n\in\mathbb N)

and the function

g(x)

associated with the graph. (b) On the graph of the function

f(x)

consider the points

M

and

M'

with

x

coordinates

x

and

f(x)

, where

x\ne c

. i. Prove that the line

MM'

intersects with the line

y=x

at the point with

x

coordinate

h(x)=x-\frac{(f(x)-x)^2}{g(x)+x-2f(x)}.

ii. Prove that if

x\in(0,c)

then

h(x)\in(x,c)

. iii. Analyze whether the sequence

\{a_n\}

satisfying

a_1\in(0,c),a_{n+1}=h(a_n)

for

n\in\mathbb N

converges. Prove that the sequence

\{\tfrac{a_{n+1}-c}{a_n-c}\}

converges and find its limit. (c) Assume that the calculator approximates every number

b\in[-2,2]

by number

\overline b

having

p

decimal digits after the decimal point. We are performing the following sequence of operations on that calculator:
1) Set

a=0.72

; 2) Calculate

\delta(a)=\overline{f(a)}-a

; 3) If

|\delta(a)|>0.5\cdot10^{-p}

, then calculate

\overline{h(a)}

and go to the operation

2)

using

\overline{h(a)}

instead of

a

; 4) If

|\delta(a)|\le0.5\cdot10^{-p}

, finish the calculation.
Let

\overleftrightarrow c

be the last of calculated values for

\overline{h(a)}

. Assuming that for each

x\in[0.72,0.73]

we have

\left|\overline{f(x)}-f(x)\right|<\epsilon

, determine

\delta(\overleftrightarrow c)

, the accuracy (depending on

\epsilon

) of the approximation of

c

with

\overleftrightarrow c

. (d) Assume that the sequence

\{a_n\}

satisfies

a_1=0.72

and

a_{n+1}=f(a_n)

for

n\in\mathbb N

. Find the smallest

n_0\in\mathbb N

, such that for every

n\ge n_0

we have

|a_n-c|<10^{-6}

Problem Statement