1.
The following diagram shows the graph of f(x) = e  x .
2
The points A, B, C, D and E lie on the graph of f. Two of these are points of inflexion.
(a)
Identify the two points of inflexion.
(2)
(b)
(i)
Find f′(x).
(ii)
Show that f″(x) = (4x2 – 2) e  x .
2
(5)
(c)
Find the x-coordinate of each point of inflexion.
(4)
(d)
Use the second derivative to show that one of these points is a point of inflexion.
(4)
(Total 15 marks)
IB Questionbank Maths SL
1
2.
Consider the function h : x 
x–2
( x – 1) 2
, x  1.
A sketch of part of the graph of h is given below.
A
y
P
x
Not to scale
B
The line (AB) is a vertical asymptote. The point P is a point of inflexion.
(a)
Write down the equation of the vertical asymptote.
(1)
(b)
Find h′(x), writing your answer in the form
a–x
( x – 1) n
where a and n are constants to be determined.
(4)
(c)
Given that h  ( x) 
2x – 8
( x – 1) 4
, calculate the coordinates of P.
(3)
(Total 8 marks)
IB Questionbank Maths SL
2
ANSWERS:
1.
(a)
(b)
B, D
(i)
f′(x) =  2 xe  x
2
Note: Award A1 for e
(ii)
N2
A1A1
N2
(A1)
evidence of choosing the product rule
(M1)
2
 2e  x  4 x 2 e  x
2
2
2
f ′′(x) = (4x2 – 2) e  x
2
and A1 for –2x.
finding the derivative of –2x, i.e. –2
e.g.  2e  x  2 x  2 xe  x
(c)
 x2
A1A1
A1
2
AG
valid reasoning
N0
5
N3
4
N0
4
R1
e.g. f ′′(x) = 0
attempting to solve the equation
(M1)
e.g. (4x2 – 2) = 0, sketch of f ′′(x)
 1 
 , q  0.707
p = 0.707  
2

(d)

1 
  

2

evidence of using second derivative to test values on either side of POI
A1A1
M1
e.g. finding values, reference to graph of f′′, sign table
correct working
A1A1
e.g. finding any two correct values either side of POI,
checking sign of f ′′ on either side of POI
reference to sign change of f ′′(x)
R1
[15]
2.
(a)
x=1
(b)
Using quotient rule
(A1)
1
(M1)
( x  1) (1)  ( x  2)[2( x  1)]
( x  1) 4
2
Substituting correctly g(x) =
( x  1)  (2 x  4)
( x  1) 3
= 3  x 3 (Accept a = 3, n = 3)
( x  1)
=
(c)
A1
(A1)
A1
Recognizing at point of inflexion g(x) = 0
x=4
M1
A1
Finding corresponding y-value = 2 = 0.222 ie P  4, 2 
9
 9
A1
4
3
[8]
IB Questionbank Maths SL
3