LLG
Advanced Math and Science Pilot Class
Paris – Abu Dhabi
Mathematics, Grade 12
2014 – 2015
Chapter 6 : Continuous function Worksheet 2
Exercise 4 : f is the function defined on ℝ by 𝑓(𝑥) = 1 + 3𝑥 − 𝑥 3 .
1. Draw the variation table of f .
2. Prove that the equation
f x   0 has a unique solution in each of the following
intervals : [−2 ; −1] ; [−1 ; 1] ; [1 ; 2].
Exercise 5 : f is a continuous function on  3 ; 2
 with the following variation table:
Exercise 1 : Here is the graph of a function f defined on [6 ; 8 ] .
Determine the images by f of the following intervals : [−2 ; 1] ; [−2 ; 2] ; [−3 ; 2].
Exercise 6 : f is a continuous function with the following variation table:
1. Is f continuous at :
(a) −4
(b) 1
2. Is f continuous on the interval :
(a) [−4 ; 0]
(b) [0 ; 2]
(c) 2 ?
(c) [2 ; 4]

1 x2 1
 f x  
if x  0
Exercise 2 : f is the function defined by: 
x
 f 0   m

1. What is the domain of f ?
2. Determine the real number m so that f is continuous at 0.
Exercise 3 :
Determine the images by f of the intervals: ]−∞ ; 0[ ; ]0 ; 2[ ; ]−∞ ; 2[ ; ]2 ; +∞[ .
Exercise 7 : Prove that the equation (𝐸): 𝑥 √𝑥 = 1 − 𝑥 has a unique solution in ℝ+ .
Exercise 8 : 1. Using the graphs below, say in each case whether the function is a bijection
from 𝐸 to 𝐹 or not :
f and g are the functions defined on ℝ by :
 x2 1
if x  1

f x    x  1
0
if x  1

 x2 1
if x  1

g x    x  1
.
m
if x  1

1. Is f continuous on ℝ ?
2. Determine the real number m so that g is continuous at -1.
(a) 𝐸 = ℝ∗ , 𝐹 = ℝ∗
(b) 𝐸 = ℝ, 𝐹 = ℝ
Exercise 11 :
1. Let 𝑓 be the function defined on ℝ by 𝑓(𝑥) = (2 − 𝑥)𝑒 𝑥 − 1.
(a) Calculate the limits of 𝑓 at the endpoints of its domain.
(b) Evaluate 𝑓 ′ and deduce the table of variations of 𝑓.
(c) Prove that 𝑓(𝑥) = 0 for two real numbers 𝛼 and 𝛽 (with 𝛼 < 𝛽). Find a
double inequality for 𝛼 and 𝛽 of width 10−2 .
(d) Give the table of signs of 𝑓.
(e) Show that 𝑒 𝛼 =
∗
(c) 𝐸 = ℝ , 𝐹 = ℝ
∗
∗
(d) 𝐸 = ℝ , 𝐹 = ℝ
For each following function, say whether it is a bijection from 𝐸 to 𝐹 or not. In cas
it is a bijection, give its inverse function :
(a) 𝑓(𝑥) = −4𝑥 + 5,
𝐸 = ℝ,
𝐹=ℝ
(b) 𝑓(𝑥) = 𝑥 2 ,
𝐸 = ℝ,
𝐹 = ℝ+
(c) 𝑓(𝑥) = 𝑥 2 ,
𝐸 = ℝ+ ,
𝐹 = ℝ+
3+2𝑥
𝑥−1
, find out 𝐸 and 𝐹 so that 𝑓is a bijection.
Exercise 9 : Let 𝑓 be the function defined on ℝ by 𝑓(𝑥) = 𝑥 2 − 𝑥 + 2𝑒 𝑥 .
1. Calculate the limits of 𝑓 at the endpoints of its domain.
2. Calculate 𝑓 ′ and 𝑓′′.
3. Give the sign of 𝑓′′ and deduce the variations of 𝑓′.
4. Prove that the equation 𝑓 ′ (𝑥) = 0 has a unique solution𝛼 in ℝ. Round it at 10−2 .
5. Give the sign of 𝑓′ and deduce the variations of 𝑓.
Exercise 10 : Let 𝑔 be the function defined on ℝ by 𝑔(𝑥) = (𝑥 − 1)𝑒 𝑥 − 1.
1. (a) Evaluate the derivative function of 𝑔.
(b) Calculate the limits of 𝑔 at the endpoints of its domain.
(c) Give the variations of 𝑔 and draw its table of variations.
(d) Prove that the equation 𝑔(𝑥) = 0 has a unique solution𝛼 in ℝ. Round it at 10−2 .
(e) Deduce the sign of 𝑔 on ℝ.
2.
Let 𝑓 be the function defined on ℝ by 𝑓(𝑥) =
𝑥
.
𝑒 𝑥 +1
(a) Calculate the limits of 𝑓 at the endpoints of its domain. Deduce that 𝐶𝑓 has a
vertical asymptote.
(b) Prove that the line with equation 𝑦 = 𝑥 is an oblique asymptote to the curve of 𝑓
in −∞.
(c) Calculate 𝑓′.
(d) Give the sign of 𝑓′ and deduce the variations of 𝑓.
.
2.
Let ℎ be the function defined by ℎ(𝑥) = 𝑒 𝑥 − 𝑥.
(a) Calculate the derivative function of ℎ.
(b) Show that for all 𝑥 ∈ ℝ, ℎ(𝑥) > 0.
3.
Let 𝑔 be the function defined by 𝑔(𝑥) =
2.
(d) 𝑓(𝑥) =
1
2−𝛼
𝑒 𝑥 −1
𝑒 𝑥 −𝑥
.
(a) Show that 𝑔 is defined on ℝ.
(b) Calculate the limits of 𝑓 at the endpoints of its domain and deduce
asymptotes if relevant.
(c) Evaluate the derivative function of 𝑔.
(d) Justify the variations of 𝑔 and draw its table of variations.
Exercise 12 :
A- An auxiliary function : We consider 𝑔 defined on ℝ by 𝑔(𝑥) = 2𝑒 𝑥 + 2𝑥 − 7.
1. Calculate the limits of 𝑔 at the endpoints of its domain.
2. Give the variations of 𝑔 and draw its table of variations.
3. Prove that the equation 𝑔(𝑥) = 0 has a unique solution 𝛼 and explain why
0.94 < 𝛼 < 0.941. Give the sign of 𝑔 on ℝ.
B- The function : We consider f defined on ℝ by 𝑓(𝑥) = (2𝑥 − 5)(1 − 𝑒 −𝑥 ) and 𝐶𝑓 its
curve in an orthonormal frame of the plane.
1. Give the sign of 𝑓 on ℝ.
2. Calculate the limits of 𝑓 at the endpoints of its domain.
3. Show that the line with equation 𝑦 = 2𝑥 − 5 is an oblique asymptote to 𝐶𝑓 . Give
its position compared to 𝐶𝑓 .
4. Evaluate 𝑓′ and check that 𝑓 ′ has the same sign as 𝑔. Deduce the variations of 𝑓.
5. (a) Show that 𝑓(𝛼) =
(2𝛼−5)2
2𝛼−7
.
(b) Give the variations of ℎ: 𝑥 ↦
(2𝑥−5)2
2𝑥−7
5
on the interval ]−∞ ; [.
2
(c) Using question A3, deduce an inequality of 𝑓(𝛼) of width 10−2 .