MCV4U1-UNIT FOUR Lesson Four
Lesson Four: Concavity and Points of Inflection
From Lesson Two, we learned:
if f ( x)  0 then f (x) is increasing, and
if f ( x)  0 then f (x) is decreasing.
How can we use derivatives to help us determine whether the graph of a function has a
hill or a valley?
Investigation:
1. Consider two functions: f ( x)  x 2  6 x  2
and
g ( x)   x 2  4 x  3
We say that the graph of f (x) has a valley and the graph of g (x) has a hill. Complete
the following tables to help find a pattern which can help us predict whether a graph will
have a hill (concave down) or a valley (concave up):
f ( x)  x 2  6 x  2
f (x) 
f (x) 
f (x )
x
f (x)
-2
-1
0
1
2
3
4
5
f (x)
f ( x)   x 2  4 x  3
f (x) 
f (x) 
f (x )
x
f (x)
-2
-1
0
1
2
3
4
5
Repeat the investigation for the two functions shown on the next page:
f (x)
MCV4U1-UNIT FOUR Lesson Four
2. Consider two functions: f ( x)  x3 12x and
g(x)  x3  12x 2  36x  5
Complete the following tables to help find a pattern which can help us predict whether a
graph will have a hill (concave down) or a valley (concave up):
f ( x)  x3 12x
f (x) 
f (x) 
f (x )
x
f (x)
-5
-4
-3
-2
-1
0
1
2
3
4
5
f (x)
f ( x)  x3  12x 2  36x  5
f (x) 
f (x) 
x
f (x )
f (x)
f (x)
-2
-1
0
1
2
3
4
5
6
7
8
Look carefully at the charts above. Can you find the determining factor as to whether the
graph of a function will have a hill or a valley?
Conclusion: Fill in the blanks:
if f (x) ____________0
if f (x)____________0
then the graph has a valley
then the graph has a hill .
ex. 1. Determine when the graph of f ( x)  3x 2  12 will have a hill or a valley:
Solution:

f ( x
)  3 x 2  12
 f ( x)  0 for all values of x, the graph always has a valley.
f ( x )  6 x
f ( x )  6
MCV4U1-UNIT FOUR Lesson Four
ex. 2. Determine when the graph of y  x 3  12x 2 has a hill:
Solution:
d2y
3
2
So the graph of y  x 3  12x 2 has
y  x  12 x
0
2
dx
a hill for all values of x < -4
dy
2
 3 x  24 x For a hill, 6 x  24  0 .
dx
x  4
d2y
 6 x  24
dx 2
An inflection point is a point on a graph where the graph changes from a hill to a valley,
or from a valley to a hill.
Conditions for an inflection point:
d2y
d2y
 0 or is undefined, and
changes sign
dx 2
dx 2

ex. 3, Find where there are inflection points for the following functions:
a) f ( x)  x 3  3x
b) y   x 3  3x 2  2
c) f ( x)  x3  3x 2  9x  11
d) y  x 4  4x3  48x 2  500
ex. 4. Explain why the graph of f ( x)  x 4 does NOT have an inflection point at x = 0.