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4
3
2
1
Outside the high school program
/ Second-degree function: study with the canonical form
38 exercises (including 30 corrected)
a
Calculator and second degree
(2 exercices)
Second degree: variation
(3 exercices)
Graphical representation
(3 exercices)
Extrema
(4 exercices)
Sense of variation
(3 exercices)
Algebraic studies
(2 exercices)
Table of variations and image of intervals
(2 exercices)
Graphic and algebraic study
(3 exercices)
Problems
(1 exercice)
Modeling
(6 exercices)
Problems and signs
(2 exercices)
Calculator and polynomial
(7 exercices)
image PNG : /home/_math/_exercice/d4/4641/metapost/a3.png
image PNG : /home/_math/_exercice/d4/4635/metapost/a1.png
image PNG : /home/_math/_exercice/d4/4634/metapost/a.png
image PNG : /home/_math/_exercice/d4/4636/metapost/a2.png
−∞
∞
−
b
2
a
x
a
·
x
2
b
·
x
c
a>
0
−∞
∞
−
b
2
a
x
a
·
x
2
b
·
x
c
a<
0
x
-5
-4
-3
-2
-1
0
1
2
y
-2
-1
1
2
3
4
1.
Calculator
and
se
c
ond
de
gr
e
e
E.4641
1
Consider
the
p
olynomial
x
3
+
x
2
−
3
x
−
3
whose
repre-
sen
tation
is
giv
en
opp
osite
:
a
The
calculator
displa
ys
an
in
teger
ro
ot
of
this
p
olyno-
mial.
Deduce
a
factorization
of
this
p
olynomial
of
the
form
:
x
3
+
x
2
−
3
x
−
3
=
x
−
¸
˛
·
x
2
+
‚
·
x
+
‹
b
F
actor
this
p
olynomial
in
to
a
pro
duct
of
first-degree
factors.
2
F
ollo
wing
the
reasoning
in
the
previous
question,
factor
the
follo
wing
p
olynomials
in
to
first-degree
factors
:
a
2
x
3
−
2
x
2
−
x
+
1
b
x
3
+
2
x
2
−
2
x
−
4
c
2
x
3
+
3
x
2
−
4
x
+
1
f
−
x
3
−
4
x
2
−
3
x
+
2
g
−
2
x
3
−
2
x
2
+
3
x
+
1
h
x
3
+
3
x
2
−
2
2.
Se
c
ond
de
gr
e
e
:
variation
E.4468
Reminders
:
A
p
olynomial
of
the
second
degree
admits
as
its
reduced
dev
elop
ed
form
a
·
x
2
+
b
·
x
+
c
where
a
=
0
.
The
function
x
↦−
→
a
·
x
2
+
b
·
x
+
c
has
its
table
of
v
ariations
de-
p
ending
on
the
sign
of
the
co
efficien
t
of
the
second-degree
term
:
Its
represen
tativ
e
curv
e
is
called
a
parab
ola
and
its
v
ertex
has
abscissa
−
b
2
a
.
Dra
w
up
the
table
of
v
ariations
for
eac
h
of
the
follo
wing
p
oly-
nomials
:
a
4
x
2
+
4
x
+
5
b
−
2
x
2
−
2
x
+
3
c
2
x
2
+
4
x
+
2
d
−
4
x
2
+
4
x
+
3
e
2
x
2
+
3
x
+
1
f
−
4
x
2
−
4
x
−
1
E.4501
1
Dra
w
up
the
table
of
v
ariations
of
the
follo
wing
func-
tions
:
a
f
(
x
)
=
x
2
+
x
−
2
b
g
(
x
)
=
−
2
x
2
+
4
x
−
3
c
h
(
x
)
=
−
4
x
2
+
x
+
2
d
j
(
x
)
=
2
x
2
+
2
x
+
2
2
F
or
eac
h
function
from
the
previous
question,
giv
e,
with-
out
sp
ecifying
their
v
alues,
the
n
um
b
er
of
an
teceden
ts
of
0
.
E.4976
F
or
eac
h
function,
determine
the
c
haracteristics
of
their
extrema
and
dra
w
up
the
table
of
v
ari-
ations
:
a
f
:
x
↦−
→
2
x
2
+
8
x
+
1
b
g
:
x
↦−
→
−
x
2
+
2
x
+
1
c
h
:
x
↦−
→
x
2
+
4
x
+
4
d
j
:
x
↦−
→
−
3
x
2
+
9
x
−
2
e
k
:
x
↦−
→
3
x
2
+
2
x
+
2
f
‘
:
x
↦−
→
−
x
2
+
2
3
x
−
1
3.
Gr
aphic
al
r
epr
esentation
E.8207
Consider
the
quadratic
function
f
defined
on
R
b
y
the
relation
:
f
(
x
)
=
0.4
x
2
+
0.8
x
−
1.2
Consider
the
plane
equipp
ed
with
the
co
ordinate
system
sho
wn
b
elo
w
:
h
ttps:/
/c
hingmath.fr
sacados/4641
chapExoCorrec/4468
sacados/4468
−∞
∞
−
b
2
a
x
a
·
x
2
b
·
x
c
a>
0
−∞
∞
−
b
2
a
x
a
·
x
2
b
·
x
c
a<
0
chapExoCorrec/4501
sacados/4501
chapExoCorrec/4976
sacados/4976
chapExoCorrec/8207
sacados/8207
x
-5
-4
-3
-2
-1
0
1
2
y
-2
-1
1
2
3
4
x
-2
-1
0
1
2
3
4
y
-2
-1
1
2
-5
-4
-3
-2
-1
I
-3
-2
-1
2
3
4
5
6
J
O
Let
C
f
b
e
the
curv
e
represen
ting
the
function
f
in
the
co
or-
dinate
system
ab
o
v
e.
1
a
Using
a
calculator,
complete
the
tables
of
v
alues
b
e-
lo
w
b
y
en
tering
the
v
alues
of
the
images
rounded
to
the
nearest
ten
th
:
x
−
4.5
−
4
−
3
−
2
−
1.5
−
1
0.5
1
2
f
(
x
)
b
Plot
the
represen
tation
of
the
curv
e
C
f
.
2
Using
a
graph
:
a
Dra
w
up
a
table
of
v
ariations
for
the
function
f
.
b
Giv
e
the
nature
of
the
extrem
um
of
the
function
f
and
its
c
haracteristics.
E.8208
Consider
the
quadratic
function
f
defined
on
R
b
y
the
relation
:
f
(
x
)
=
−
0.5
x
2
+
x
+
1.5
Consider
the
plane
equipp
ed
with
the
co
ordinate
system
sho
wn
b
elo
w
:
Let
C
f
b
e
the
curv
e
represen
ting
the
function
f
in
the
co
or-
dinate
system
ab
o
v
e.
1
a
Using
a
calculator,
complete
the
tables
of
v
alues
b
e-
lo
w
b
y
en
tering
the
v
alues
of
the
images
rounded
to
the
nearest
ten
th
:
x
−
2
−
1.5
−
1
0
0.5
1
1.5
2
3
f
(
x
)
b
Plot
the
represen
tation
of
the
curv
e
C
f
.
2
Using
a
graph
:
a
Dra
w
up
a
table
of
v
ariations
for
the
function
f
.
b
Giv
e
the
nature
of
the
extrem
um
of
the
function
f
and
its
c
haracteristics.
E.1762
Consider
the
function
:
f
:
x
↦−
→
x
2
+
4
x
+1
:
1
Establish
equalit
y
:
f
(
x
)
=
(
x
+2)
2
−
3
2
a
Determine
the
direction
of
v
ariation
of
the
function
f
on
the
in
terv
al
−∞
;
−
2
b
Dra
w
up,
without
justification,
its
table
of
v
ariation.
c
Giv
e
the
c
haracteristics
of
the
extrem
um
of
the
func-
tion
f
.
3
a
Complete
the
table
b
elo
w
of
v
alues
for
the
function
f
:
x
−
5
−
4
−
3.5
−
3
−
2.5
−
2
f
(
x
)
x
−
1.5
−
1
−
0.5
0
1
f
(
x
)
b
Dra
w
the
represen
tativ
e
curv
e
C
f
of
the
function
f
in
the
frame
b
elo
w
:
4.
Extr
ema
E.8204
Consider
the
second-degree
function
f
defined
on
R
b
y
:
f
(
x
)
=
2
x
2
−
8
x
+
5
1
Establish
iden
tit
y
:
f
(
x
)
=
2(
x
−
2)
2
−
3
2
Justify
that
the
function
f
admits
for
minim
um
the
v
alue
−
3
reac
hed
for
x
=
2
.
E.8205
Consider
the
second-degree
function
f
defined
on
R
b
y
:
f
(
x
)
=
−
3
x
2
−
12
x
−
13
1
Establish
iden
tit
y
:
f
(
x
)
=
−
3(
x
+2)
2
−
1
2
Justify
that
the
function
f
admits
for
maxim
um
the
v
alue
−
1
reac
hed
for
x
=
−
2
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8208
sacados/8208
x
-2
-1
0
1
2
3
4
y
-2
-1
1
2
chapExoCorrec/1762
sacados/1762
-5
-4
-3
-2
-1
I
-3
-2
-1
2
3
4
5
6
J
O
chapExoCorrec/8204
sacados/8204
chapExoCorrec/8205
sacados/8205
E.425
Consider
the
function
f
defined
b
y
:
f
:
x
↦−
→
9
x
2
−
6
x
−
2
1
a
Giv
e
the
definition
set
of
the
function
f
.
b
Calculate
the
image
of
1
b
y
the
function
f
.
c
Determine
the
an
teceden
ts
of
−
2
b
y
the
function
f
.
2
a
Establish
equalit
y
:
9
x
2
−
6
x
−
2
=
9
x
−
1
3
2
−
3
.
b
Sho
w
that
f
admits
a
minim
um
and
that
this
is
reac
hed
in
1
3
.
E.8206
F
or
eac
h
of
the
second-degree
func-
tions
b
elo
w,
giv
e
the
nature
of
its
extrem
um
and
its
c
harac-
teristics
:
1
f
(
x
)
=
x
2
+
4
x
+
1
2
g
(
x
)
=
−
3
x
2
+
6
x
−
3
3
h
(
x
)
=
x
2
+
5
x
−
4
4
j
(
x
)
=
−
5
x
2
+
4
x
+
1
5.
Sense
of
variation
E.414
Consider
the
function
f
defined
on
R
whose
image
of
a
n
um
b
er
x
is
defined
b
y
:
f
(
x
)
=
−
9
x
2
−
12
x
+
1
1
Establish
equalit
y
:
f
(
x
)
=
−
9
x
+
2
3
2
+5
.
2
Demonstrate
that,
on
−∞
;
−
2
3
,
the
function
f
is
in-
creasing.
E.1760
Consider
the
function
f
whose
image
of
a
real
n
um
b
er
x
is
defined
b
y
the
relation
:
f
(
x
)
=
2
x
2
+
4
x
−
1
1
Establish
equalit
y
:
f
(
x
)
=
2(
x
+1)
2
−
3
2
Sho
w
that
:
a
f
is
strictly
decreasing
on
−∞
;
−
1
.
b
f
is
strictly
increasing
on
−
1;+
∞
.
3
Dra
w
up
the
table
of
v
ariations
of
the
function
f
.
4
Deduce
that
−
3
is
the
minim
um
of
the
function
f
.
E.1753
Consider
the
function
f
defined
on
R
suc
h
that
the
image
of
a
n
um
b
er
x
is
giv
en
b
y
the
relation
:
f
(
x
)
=
−
2
x
2
−
8
x
+
9
.
1
a
Establish
equalit
y
:
f
(
x
)
=
−
2
·
(
x
+2)
2
+17
.
b
Establish
the
direction
of
v
ariation
of
the
function
f
on
the
in
terv
al
−∞
;
−
2
.
2
Dra
w
up,
without
justification,
the
table
of
v
ariations
of
the
function
f
.
3
Giv
e
the
c
haracteristics
of
the
extrem
um
of
the
function
f
.
6.
A
lgebr
aic
studies
E.406
Consider
the
function
f
defined
on
R
b
y
:
f
:
x
↦−
→
−
2
x
2
+
2
x
+
12
1
Establish
the
follo
wing
equalities
:
−
2
x
2
+
2
x
+
12
=
(3
−
x
)(2
x
+
4)
=
−
2
x
−
1
2
2
+
25
2
2
a
Justify
that
the
function
f
cancels
for
t
w
o
n
um
b
ers
to
b
e
sp
ecified.
b
Justify
that
the
function
f
admits
25
2
as
its
maxim
um.
3
a
Establish
that
the
function
f
is
increasing
on
the
in
terv
al
−∞
;
1
2
.
b
Dra
w
up,
without
justification,
the
table
of
v
ariations
of
the
function
f
on
R
.
E.1757
Consider
the
function
f
defined
on
R
whose
image
of
a
n
um
b
er
x
is
defined
b
y
the
relation
:
f
(
x
)
=
8
x
2
−
8
x
−
6
.
1
a
Establish
iden
tit
y
:
f
(
x
)
=
4
x
−
6
2
x
+
1
b
Dra
w
up
the
sign
table
for
the
function
f
.
2
a
Establish
the
iden
tit
y
:
f
(
x
)
=
8
x
−
1
2
2
−
8
.
b
Algebraically
,
establish
the
follo
wing
inequalit
y
for
an
y
real
n
um
b
er
x
:
f
(
x
)
−
8
.
c
On
the
in
terv
al
−∞
;
1
2
,
establish
the
direction
of
v
ariation
of
f
3
a
Dra
w
up,
without
justification,
a
table
of
v
ariations
of
the
function
f
.
b
Giv
e,
without
justification,
the
c
haracteristics
of
the
extrem
um
of
the
function
f
.
7.
T
able
of
variations
and
image
of
intervals
h
ttps:/
/c
hingmath.fr
chapExoCorrec/425
sacados/425
chapExoCorrec/8206
sacados/8206
chapExoCorrec/414
sacados/414
chapExoCorrec/1760
sacados/1760
chapExoCorrec/1753
sacados/1753
chapExoCorrec/406
sacados/406
chapExoCorrec/1757
sacados/1757
-4
-3
-2
-1
2
I
-2
-1
2
J
O
C
f
E.8209
1
Consider
the
function
f
defined
on
R
b
y
:
f
(
x
)
=
2
x
2
−
x
+
1
a
Dra
w
up
the
table
of
v
ariations
of
the
function
f
.
b
Deduce
the
images
of
the
follo
wing
in
terv
als
b
y
the
function
f
:
A
=
−
3
;
1
4
;
B
=
1
;
2
;
C
=
−
3
;
1
2
Consider
the
function
g
defined
on
R
b
y
:
g
(
x
)
=
−
x
2
+
2
x
−
4
a
Dra
w
up
the
table
of
v
ariations
of
the
function
g
.
b
Deduce
the
images
of
the
follo
wing
in
terv
als
b
y
the
function
g
:
D
=
−
3
;
0
;
E
=
1
;
3
;
G
=
0
;
3
E.405
Consider
the
function
f
defined
b
y
the
relation
:
f
(
x
)
=
4
x
2
−
16
x
+
9
.
1
Establish
equalit
y
:
4
x
2
−
16
x
+9
=
4(
x
−
2)
2
−
7
2
Sho
w
that
:
a
function
f
is
decreasing
on
−∞
;
2
.
b
function
f
is
increasing
on
2
;
+
∞
.
3
Dra
w
up
the
table
of
v
ariations
of
the
function
f
.
4
Determine
the
image
of
the
follo
wing
in
terv
als
b
y
the
function
f
:
a
−
2
;
1
b
0
;
4
c
1
;
+
∞
8.
Gr
aphic
and
algebr
aic
study
E.437
Consider
the
function
f
,
whic
h
to
an
y
n
um
b
er
x
,
asso
ciates
its
image
f
(
x
)
defined
b
y
:
f
(
x
)
=
1
2
x
2
+
x
−
3
2
The
graph
b
elo
w
sho
ws
the
represen
tation
of
the
C
f
curv
e
in
the
O
;
I
;
J
orthonormal
co
ordinate
system
:
P
art
A
:
graphical
study
Graphically
,
answ
er
the
follo
wing
questions
:
1
Giv
e
the
an
teceden
ts
of
the
n
um
b
er
0
b
y
f
.
2
Dra
w
up
the
table
of
v
ariations
of
the
function
f
on
the
in
terv
al
−
4
;
2
.
3
Giv
e
the
c
haracteristics
of
the
extrem
um
of
the
function
f
.
P
art
B
:
algebraic
study
1
a
Establish
the
follo
wing
equalit
y
:
f
(
x
)
=
1
2
·
x
+
3
x
−
1
b
Solv
e
the
equation
:
f
(
x
)
=
0
.
2
a
Establish
the
follo
wing
equalit
y
:
f
(
x
)
=
1
2
·
x
+
1
2
−
2
b
Demonstrate
that
the
function
f
is
decreasing
on
the
in
terv
al
−∞
;
−
1
.
E.4869
Consider
the
function
f
defined
b
y
the
expression
:
f
(
x
)
=
a
·
x
2
+
b
·
x
+
c
where
a
,
b
,
c
are
real
n
um
b
ers.
The
parab
ola
C
f
represen
tativ
e
of
the
function
f
has
as
its
v
ertex
the
p
oin
t
with
co
ordinates
(
−
1
;
2
)
and
passes
through
the
p
oin
t
with
co
ordinates
(
−
3
;
0
)
.
Determine,
without
justification,
the
v
alues
of
the
real
n
um-
b
ers
a
,
b
and
c
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8209
sacados/8209
chapExoCorrec/405
sacados/405
chapExoCorrec/437
sacados/437
-4
-3
-2
-1
2
I
-2
-1
2
J
O
C
f
chapExoCorrec/4869
sacados/4869
-2
-1
2
3
I
-4
-2
2
J
O
A
B
C
D
E
2
m
y
3
m
x
3
y
x
A
B
C
D
E.6689
The
curv
e
opp
osite
is
the
represen
tation
of
a
function
f
defined
on
−
2
;
3
.
1
Graphically
,
answ
er
the
follo
wing
questions
:
a
What
are
the
images
of
0
and
2
b
y
f
?
b
Giv
e,
if
p
ossible
:
an
y
history
of
−
4
b
y
f
;
p
ossible
history
of
4
b
y
f
;
c
What
are
the
solutions
to
the
equation
:
f
(
x
)
=
−
7
4
?
d
What
are
the
solutions
to
the
inequation
:
f
(
x
)
<
0
?
2
W
e
admit
that
the
function
f
is
defined
on
R
and
admits
for
expression
:
f
(
x
)
=
−
x
2
+
x
+
2
.
a
Justify
that
:
f
(
x
)
=
(
−
x
+2)(
x
+1)
b
Solv
e
the
equation
:
f
(
x
)
=
0
c
Using
a
sign
table,
solv
e
the
inequation
f
(
x
)
0
.
d
Justify
that
:
f
(
x
)
=
−
x
−
1
2
2
+
9
4
.
Deduce
the
gro
wth
of
f
on
the
in
terv
al
−
2
;
1
2
.
9.
Pr
oblems
E.4878
Consider
the
figure
b
elo
w
:
It
v
erifies
the
follo
wing
conditions
:
The
triangle
AB
C
is
rectangular
at
A
suc
h
that
:
AB
=
3
m
;
AC
=
2
cm
The
p
oin
t
D
is
defined
b
y
:
D
∈
[
AB
)
;
D
∈
[
AB
]
p
oin
t
E
is
defined
b
y
:
E
∈
[
AC
)
;
E
∈
[
AC
]
W
e
ha
v
e
the
relationship
:
B
D
+
C
E
=
10
m
.
Let
x
and
y
b
e
the
resp
ectiv
e
lengths
of
segmen
ts
[
B
D
]
and
[
C
E
]
.
1
a
Establish
the
iden
tit
y
:
2
x
2
−
18
x
+
153
=
2
x
−
9
2
2
+
225
2
b
Determine
the
v
alues
of
x
and
y
so
that
the
length
of
the
segmen
t
[
D
E
]
is
minimal.
2
Determine
the
p
ossible
v
alues
of
x
and
y
so
that
the
area
of
the
hatc
hed
part
measures
15
m
2
10.
Mo
deling
E.2865
W
e
w
an
t
to
construct
a
rectangular
pla
y
area
along
the
side
of
a
building.
F
urthermore,
w
e
w
an
t
the
dimensions
of
this
rectangle
to
b
e
greater
than
or
equal
to
10
m
.
This
pla
ying
area
is
surrounded
on
three
sides
b
y
a
3
m
wide
alley
as
sho
wn
in
the
sk
etc
h
b
elo
w.
The
set
is
fenced
on
three
sides
[
AB
]
,
[
B
C
]
and
[
C
D
]
.
W
e
are
in
terested
in
the
length
L
of
the
fence
:
L
=
AB
+
B
C
+
C
D
.
Let
x
and
y
b
e
the
dimensions
in
meters
of
the
pla
ying
area
(the
value
of
x
and
y
ar
e
ne
c
essarily
p
ositive)
.
1
Express
the
length
L
of
the
fence
in
terms
of
the
v
alues
of
x
and
y
.
2
W
e
ha
v
e
100
meters
of
fence
that
w
e
w
an
t
to
use
en
tirely
:
a
Express,
under
these
conditions,
the
v
alue
of
y
as
a
function
of
x
.
b
Justify
that
the
v
alue
of
x
m
ust
b
e
less
than
44.
c
Justify
that
the
area
of
the
pla
yground
is
:
A
(
x
)
=
88
x
−
2
x
2
3
a
Justify
the
equalit
y
b
elo
w
:
A
(
x
)
=
968
−
2(
x
−
22)
2
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6689
sacados/6689
-2
-1
2
3
I
-4
-2
2
J
O
chapExoCorrec/4878
sacados/4878
fichierPlus/4878/
A
B
C
D
E
2
m
y
3
m
x
chapExoCorrec/2865
sacados/2865
3
y
x
A
B
C
D
xcm
y
cm
3
cm
b
Deduce
the
gro
wth
of
the
function
A
on
the
in
terv
al
0
;
22
.
c
Dra
w
up,
without
justification,
the
table
of
v
ariations
of
the
function
A
on
the
in
terv
al
0
;
44
.
4
Deduce
the
dimensions
so
that
the
100
meters
of
fencing
are
used
and
the
pla
y
area
is
maximized.
E.2877
A
carp
en
ter
has
a
stic
k
of
w
o
o
d
100
cen
timetres
long
and
3
cen
timetres
wide.
He
w
an
ts
to
use
the
full
length
of
this
stic
k
to
mak
e
a
w
o
o
den
frame
lik
e
the
one
sho
wn
in
the
dra
wing
b
elo
w
:
1
a
Determine
the
v
alue
of
y
as
a
function
of
x
.
b
Deduce
the
p
ossible
v
alues
of
x
.
2
Giv
e
the
expression
for
the
area
A
(
x
)
of
the
inside
of
the
frame
as
a
function
of
x
.
3
a
Establish
the
follo
wing
equalit
y
:
A
(
x
)
=
−
(
x
−
28)
2
+
484
b
Establish
the
gro
wth
of
the
function
A
on
the
in
terv
al
6
;
28
.
c
Dra
w
up
the
table
of
v
ariations
of
the
function
A
,
then
giv
e
the
maxim
um
area
reac
hed
b
y
the
frame.
E.3024
A
t
a
funfair,
the
manager
of
the
attraction
ˇ
the
crazy
caterpillar
ı
mak
es
the
follo
wing
obser-
v
ation
:
Its
carousel
can
accommo
date
70
p
eople
p
er
ride
;
If
he
sets
the
price
at
1
e
,
his
ride
is
full
ev
ery
round
;
Ev
ery
time
he
increases
the
price
b
y
0.50
e
,
he
loses
5
customers.
Let
x
b
e
the
price
of
a
seat
:
the
n
um
b
er
x
b
elongs
to
the
in
terv
al
1
;
+
∞
[
.
W
e
are
in
terested
in
the
v
alue
of
the
rev
en
ue
made
b
y
this
at-
traction
eac
h
round
as
a
function
of
the
price
x
of
the
seats
;
w
e
note
R
(
x
)
the
v
alue
of
this
rev
en
ue.
1
Justify
that
the
recip
e
admits
the
expression
:
R
(
x
)
=
80
x
−
10
x
2
2
a
Dra
w
up
the
table
of
v
ariations
of
the
function
R
.
b
Deduce
the
maxim
um
rev
en
ue
this
attraction
can
ac
hiev
e.
What
m
ust
the
price
of
a
ride
b
e
to
ac
hiev
e
this
maxim
um?
3
The
manager
w
an
ts
to
earn
more
than
150
e
à
for
eac
h
ride.
Let’s
determine
the
p
ossible
prices
of
a
seat
meeting
this
condition
:
a
Expand
the
expression
:
(
x
−
4)
2
−
1
.
Deduce
the
canonical
form
of
the
expression
R
(
x
)
−
150
.
b
F
actor
the
expression
:
(
x
−
4)
2
−
1
2
.
Then
factor
the
expression
R
(
x
)
−
150
.
c
Determine
the
solutions
of
the
equation
:
R
(
x
)
=
150
.
d
Using
the
previous
question
and
the
table
of
v
ariations
of
the
function
R
,
giv
e
without
justification
the
set
of
solutions
of
the
inequation
R
(
x
)
150
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2877
sacados/2877
xcm
y
cm
3
cm
chapExoCorrec/3024
sacados/3024
A
B
C
M
N
P
Q
O
2
3
4
5
6
7
8
I
2
3
4
5
J
O
A
B
S
B
M
R
O
8
k
m
−
i
−
j
E.4852
Consider
the
isosceles
triangle
AB
C
of
dimensions
:
O
A
=
8
cm
;
O
C
=
5
cm
[
C
O
]
represen
ts
the
heigh
t
of
the
triangle
AB
C
originating
from
the
v
ertex
C
.
W
e
wish
to
inscrib
e
a
rectangle
M
N
P
Q
inside
the
triangle
and
cen
tered
around
the
axis
(
O
C
)
as
sho
wn
ab
o
v
e.
Note
x
the
length
of
segmen
t
[
O
P
]
.
1
Briefly
justify
the
p
ossible
v
alues
tak
en
b
y
the
v
ariable
x
.
2
Determine
the
measure
of
segmen
t
[
N
P
]
as
a
function
of
length
x
.
3
Sho
w
that
the
area
A
of
the
rectangle
M
N
P
Q
is
ex-
pressed
as
a
function
of
x
:
A
(
x
)
=
−
5
4
·
x
2
+
10
·
x
4
F
or
what
v
alue
is
the
area
of
the
rectangle
M
N
P
Q
max-
im
um?
E.3078
A
bask
etball
pla
y
er
at
p
oin
t
A
sho
ots
a
free
thro
w
to
w
ards
the
bask
et
at
B
;
the
ball’s
tra
jectory
passes
through
a
v
ertex
named
S
.
Jo
e,
the
mathematician
sitting
in
the
bleac
hers,
uses
a
O
;
I
;
J
orthonormal
reference
frame
to
mo
del
this
tra
jec-
tory
:
He
notes
the
co
ordinates
of
the
follo
wing
p
oin
ts
:
A
(
1
;
1
)
;
B
(
7
;
2
)
;
S
(
4.5
;
y
S
)
(Jo
e
didn
’t
have
time
to
pick
up
the
y-inter
c
ept
of
the
S
vertex)
Neglecting
friction
in
the
air,
Jo
e
kno
ws
that
an
y
pro
jectile
describ
es
a
parab
olic
tra
jectory
whose
Cartesian
equation
is
of
the
form
:
(
E
)
:
y
=
a
·
x
2
+
b
·
x
+
c
1
a
Using
the
co
ordinates
of
v
ertex
S
,
justify
the
follo
w-
ing
equalit
y
:
b
=
−
9
a
b
Justify
that
the
real
n
um
b
ers
a
and
b
v
erify
the
system
of
equations
:
c
−
8
a
=
1
c
−
14
a
=
2
c
Deduce
the
equation
(
E
)
represen
ting
the
curv
e
C
.
2
A
second
pla
y
er
mak
es
his
thro
w
;
Jo
e
obtains
the
equa-
tion
represen
ting
the
second
tra
jectory
:
(
E
)
:
y
=
−
1
8
·
x
2
+
7
6
·
x
−
1
24
a
Determine
the
v
alues
of
a
and
b
v
erifying
the
equalit
y
b
elo
w
:
−
3
x
2
+
28
x
−
49
=
x
−
7
a
·
x
+
b
b
Deduce
the
abscissas
where
the
ball
will
b
e
at
a
heigh
t
of
2
m
.
E.2988
A
b
oat
has
to
na
vigate
b
et
w
een
the
shore
and
a
ro
c
k.
This
ro
c
k
is
8
k
m
from
the
shore.
F
or
safet
y
reasons,
the
captain
wishes
to
k
eep
an
equal
dis-
tance
from
the
ro
c
k
and
the
shore.
T
o
mo
del
this
problem,
w
e’ll
use
the
reference
frame
O
;
−
→
i
;
−
→
j
whose
p
oin
t
O
is
the
orthogonal
pro
ject
of
p
oin
t
R
on
to
the
shoreline.
Note
(
x
;
y
)
the
co
ordinates
of
the
b
oat
in
this
frame
of
refer-
ence.
1
Express
as
a
function
of
x
and
y
the
distance
separating
the
b
oat
:
a
du
ro
c
her
b
du
riv
age
2
a
Using
the
equalit
y
B
R
=
B
M
,
express
the
v
alue
of
y
as
a
function
of
x
.
b
What
is
the
name
of
the
b
oat’s
tra
jectory?
11.
Pr
oblems
and
signs
E.2985
In
the
plane,
consider
a
rectangle
AB
C
D
suc
h
that
:
D
C
=
7
cm
;
D
A
=
5
cm
The
p
oin
ts
I
,
J
,
K
,
L
are
p
oin
ts
b
elonging
resp
ectiv
ely
to
the
sides
[
AB
]
,
[
B
C
]
,
[
C
D
]
,
[
D
A
]
v
erifying
the
relations
:
D
K
=
C
J
=
B
I
=
LA
=
x
cm
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4852
sacados/4852
A
B
C
M
N
P
Q
O
chapExoCorrec/3078
sacados/3078
2
3
4
5
6
7
8
I
2
3
4
5
J
O
A
B
S
chapExoCorrec/2988
sacados/2988
B
M
R
O
8
k
m
−
i
−
j
chapExoCorrec/2985
sacados/2985
A
B
C
D
I
J
K
L
xcm
A
B
M
C
C
C
1
Justify
,
briefly
,
that
the
quadrilateral
I
J
K
L
is
a
paral-
lelogram.
2
a
Determine
the
p
ossible
v
alues
for
x
.
b
Justify
that
the
area
of
the
parallelogram
I
J
K
L
has
the
expression
as
a
function
of
x
:
A
(
x
)
=
2
x
2
−
12
x
+
35
c
Deduce
the
table
of
v
ariations
of
the
function
A
.
d
Giv
e
the
minim
um
of
the
v
alue
of
the
area
of
I
J
K
L
and
the
v
alue
of
x
for
whic
h
it
is
reac
hed.
3
W
e
wish
to
determine
the
v
alues
of
x
for
whic
h,
A
is
greater
than
25
cm
2
a
Determine
the
v
alues
of
a
and
b
real
n
um
b
ers
v
erifying
the
equalit
y
:
2
x
2
−
12
x
+
10
=
(
a
·
x
+
b
)(
x
−
5)
b
Deduce
the
solutions
of
the
inequation
:
A
(
x
)
25
cm
2
.
E.3003
In
the
plane,
consider
the
circle
C
of
diameter
[
AB
]
suc
h
that
AB
=
5
cm
.
Let
M
b
e
a
p
oin
t
on
the
segmen
t
[
AB
]
,
w
e
construct
the
circles
C
and
C
of
diameters
[
AM
]
and
[
M
B
]
resp
ectiv
ely
.
W
e
mark
the
p
osition
of
p
oin
t
M
b
y
the
measure
of
AM
,
whic
h
w
e
note
x
;
w
e
are
in
terested
in
the
area
of
the
hatc
hed
part,
whic
h
w
e
note
A
(
x
)
.
1
Briefly
describ
e
the
v
alues
tak
en
b
y
the
v
ariable
x
.
2
Determine
an
expression
for
A
as
a
function
of
x
.
Consider
the
function
that
to
x
asso
ciates
A
(
x
)
3
Dra
w
up
the
table
of
v
ariations
of
the
function
A
(
x
)
.
4
a
Determine
the
v
alue
of
the
reals
a
and
b
suc
h
that
:
−
2
x
2
+
10
x
−
8
=
(
x
−
4)(
a
·
x
+
b
)
b
Solv
e
the
follo
wing
inequation
:
A
(
x
)
2
·
ı
12.
Calculator
and
p
olynomial
E.4635
Consider
the
function
f
whose
im-
age
of
a
real
n
um
b
er
x
is
defined
b
y
the
relation
:
f
(
x
)
=
−
2
x
3
−
12
x
2
+
2
x
+
12
The
represen
tation
of
this
function
is
giv
en
in
the
screenshot
opp
osite
from
a
calculator.
Of
the
follo
wing
expressions,
only
one
represen
ts
the
factor-
ized
form
of
this
p
olynomial.
Whic
h
is
it?
a
2(
x
−
1)(
x
+
1)(
x
−
6)
b
2(
x
−
1)(
x
+
1)(
x
+
6)
c
2(1
−
x
)(
x
+
1)(
x
+
6)
d
2(1
−
x
)(
x
+
1)(
x
−
6)
Chec
k
y
our
answ
er.
E.4634
Consider
the
function
f
whose
im-
age
of
a
real
n
um
b
er
x
is
defined
b
y
the
relation
:
f
(
x
)
=
x
3
+
3
x
2
−
6
x
−
8
The
represen
tation
of
this
function
is
sho
wn
in
the
screenshot
of
a
calculator
opp
osite.
Among
the
follo
wing
expressions,
only
one
represen
ts
the
fac-
tored
form
of
this
p
olynomial.
Whic
h
one?
a
(
x
+
2)(
x
−
1)(
x
−
4)
b
(
x
−
2)(
x
+
1)(
x
+
4)
c
(
x
−
1)(
x
−
2)(
x
+
4)
d
(
x
+
4)(
x
−
1)(
x
+
2)
Chec
k
y
our
answ
er.
h
ttps:/
/c
hingmath.fr
A
B
C
D
I
J
K
L
xcm
chapExoCorrec/3003
sacados/3003
A
B
M
C
C
C
sacados/4635
sacados/4634
16cm
x
x
10
cm
16
12
x
x
4
cm
6
cm
y
x
A
B
C
D
E
F
G
E.4636
1
Consider
the
t
w
o
functions
f
and
g
defined
b
y
:
f
(
x
)
=
−
x
3
+
x
2
+
4
x
+
2
g
(
x
)
=
−
x
−
1
Represen
tations
of
these
t
w
o
functions
are
giv
en
in
the
screenshot
opp
osite
from
a
calculator.
a
Use
y
our
calculator
to
describ
e
the
set
of
solutions
to
the
equation
:
f
(
x
)
=
g
(
x
)
b
F
or
eac
h
of
the
t
w
o
equalities
b
elo
w,
determine
the
v
alues
of
the
real
¸
,
˛
and
‚
realizing
these
equalities
:
−
x
3
+
x
2
+
5
x
+
3
=
x
+
1
¸
·
x
2
+
˛
·
x
+
‚
−
x
3
+
x
2
+
5
x
+
3
=
x
−
3
¸
·
x
2
+
˛
·
x
+
‚
c
Using
the
previous
question,
affirm
the
conjecture
in
question
a
.
2
Consider
the
t
w
o
functions
f
and
g
defined
b
y
:
f
(
x
)
=
2
x
3
−
5
x
+
6
;
g
(
x
)
=
x
+
2
F
ollo
w
the
approac
h
in
the
previous
question
to
deter-
mine
the
set
of
solutions
of
the
equation
:
f
(
x
)
=
g
(
x
)
E.4633
1
Using
the
calculator,
conjecture
the
factorized
form
of
the
follo
wing
p
olynomials
:
a
x
3
+
3
x
2
−
6
x
−
8
b
x
3
−
3
x
2
−
x
+
3
c
−
x
3
−
x
2
+
4
x
+
4
d
3
x
2
−
6
x
2
−
3
x
+
6
e
4
x
3
+
8
x
2
−
4
x
−
8
f
−
2
x
3
+
4
x
2
+
6
x
2
Algebraically
v
erify
y
our
conjectures
from
the
previous
question.
E.4639
Solv
e
the
follo
wing
inequations
:
a
x
3
−
4
x
2
+
x
+
6
<
0
b
2
x
3
+
4
x
2
−
6
x
0
c
−
2
x
3
−
4
x
2
+
2
x
+
4
>
0
d
2
x
3
+
2
x
2
−
2
x
−
2
0
E.4638
In
the
pattern
b
elo
w,
w
e
w
an
t
to
mak
e
a
rectangular
b
o
x
without
a
lid.
The
lengths
are
expressed
in
cm
.
Deduce
the
v
alue(s)
of
x
for
whic
h
this
b
o
x
has
an
area
of
144
cm
2
.
E.4637
On
a
former
w
asteland
of
rectangular
shap
e
of
length
16
m
and
12
m
,
the
m
unicipalit
y
wishes
to
build
a
kindergarten
with
a
driv
ew
a
y
running
around
the
pla
y
area
:
The
pla
yground
is
represen
ted
b
elo
w
b
y
the
hatc
hed
area
:
Quelle
(s)
dimension
(s)
can
the
width
of
the
driv
ew
a
y
b
e
so
that
the
pla
y
area
is
the
same
as
the
driv
ew
a
y?
13.
Unclassifie
d
financial
ye
ars
E.4879
Let
AB
C
D
b
e
a
rectangle
with
di-
mensions
6
cm
and
4
cm
.
Consider
the
p
oin
ts
E
and
G
,
lo-
cated
outside
the
rectangle
AB
C
D
,
b
elonging
resp
ectiv
ely
to
the
half-lines
[
AB
)
and
[
AD
)
and
the
p
oin
t
F
suc
h
that
the
quadrilateral
AGF
E
is
a
rectangle.
W
e
note
x
and
y
the
follo
wing
t
w
o
distances
:
x
=
D
G
;
y
=
B
E
P
oin
ts
E
and
G
m
ust
form
a
rectangle
AE
F
G
with
a
p
erime-
ter
of
28
cm
.
1
a
Sho
w
that
the
length
y
is
expressed
as
a
function
of
x
b
y
:
y
=
4
−
x
b
Deduce
the
p
ossible
v
alues
of
x
.
Let
A
b
e
the
area
of
the
shaded
part
(that
of
the
p
olygon
B
E
F
GD
C
)
.
2
Establish
that
the
area
of
the
shaded
region
can
b
e
writ-
ten
as
a
function
of
x
is
obtained
b
y
the
equation
:
A
(
x
)
=
−
x
2
+
2
x
+
24
3
Dra
w
up
the
table
of
v
ariations
of
the
function
A
on
R
.
(indic
ate
the
value
of
the
lo
c
al
extr
emum)
.
4
Let’s
study
the
extreme
v
alues
tak
en
b
y
the
shaded
area
of
this
figure
:
a
What
is
the
maxim
um
area
of
the
shaded
part?
F
or
h
ttps:/
/c
hingmath.fr
sacados/4636
sacados/4633
sacados/4639
sacados/4638
16cm
x
x
10
cm
sacados/4637
16
12
x
x
chapExoCorrec/4879
sacados/4879
4
cm
6
cm
y
x
A
B
C
D
E
F
G
what
v
alues
of
x
is
it
reac
hed?
b
What
is
the
minim
um
area
of
the
shaded
part?
F
or
what
v
alue
of
x
is
this
minim
um
ac
hiev
ed?
h
ttps:/
/c
hingmath.fr
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