Stationary Points
Definition – Stationary Point
• A stationary point of a function f, is the point
(c, f(c)), where c is in the domain of f, such
that f’(c) = 0.
• i.e. for y = f(x), dy  0
dx
Stationary points
• Note that a stationary point is a coordinate
pair (x, y) and all stationary points are critical
points.
Example 1
Find the stationary points of the function
f(x) = 4x3 – 3x2 + 9.
So, we must differentiate and find values of x for
which f’(x) = 0.
Example 1
f x   4 x 3  3x 2  9
f ' x   0
x0
 f ' x   12 x 2  6 x
 12 x 2  6 x  0
or
x  0  f x   9
 6 x2 x  1  0
1
x
2
and
1
35
x   f x  
2
4
Hence, the stationary points are:
0,9
and
 1 35 
 , 
2 4 
Determining the nature of a stationary
point
• If f’(x) > 0 for all x within an interval then the
function is increasing for that interval.
f(x)
f(x)
x
f(x)
x
x
Determining the nature of a stationary
point
• If f’(x) < 0 for all x within an interval then the
function is decreasing for that interval.
f(x)
f(x)
x
f(x)
x
x
Determining the nature of a maximum
stationary point
• Consider a maximum stationary point:
Maximum
Determining the nature of a maximum
stationary point
• To the left of the maximum stationary point
the function is increasing and to the right
the function is decreasing.
• Therefore, by the definition of a increasing
and decreasing function, the derivative has
changed sign from positive to negative.
f’(x) > 0
f’(x) < 0
Maximum
Determining the nature of a minimum
stationary point
• Consider a minimum stationary point:
Minimum
Determining the nature of a minimum
stationary point
• To the left of the minimum stationary point
the function is decreasing and to the right
the function is increasing.
• Therefore, by the definition of a increasing
and decreasing function, the derivative has
changed sign from negative to positive.
Minimum
f’(x) < 0
f’(x) > 0
First Derivative Test
• This process is known as the first derivative
test.
• If the sign of f’(x) changes from positive to
negative either side of the stationary point
then it is a maximum point.
• If the sign of f’(x) changes from negative to
positive either side of the stationary point
then it is a minimum point.
Example 6
Find the stationary points of y = 2x3 – 6x + 2
and determine their nature using the first
derivative test.
Solution:
Differentiate and find values of x for which y’ = 0
and then evaluate y’ either side of the stationary
point(s) to determine the nature.
Example 6
y  2x  6x  2
3
 y'  6 x  6
2


y'  0  6 x  6  0  6 x  1  0
2
 x 1  0
2
x  1  y  2
and
x  1  y  6
 x  1
2
Example 6
We must now determine the nature of the
stationary points using the first derivative test:
x
-2
-1
0
1
2
y’ = 6x2 – 6
18
0
-12
0
18
Either side of x = -1, y’ changes
from positive to negative.
Hence, (-1, 2) is a maximum point.
Either side of x = 1, y’ changes
from negative to positive.
Hence, (1, 6) is a minimum point.
First Derivative Test
• We have considered the first derivative test
when the gradient changes from positive to
negative or from negative to positive.
• Is it possible that the gradient does not
change on either side of a stationary point?
• To answer this we will consider y = x3.
y = x3
• We know that y = x3 looks like this:
y’ = 0: y’ = 3x2 = 0 obtains x = 0.
Therefore (0, 0) is a stationary point but it is
clearly not a maximum nor a minimum.
The first derivative test confirms that the
stationary point at x = 0 is neither a maximum
nor a minimum since y’ > 0 on either side of the
stationary point. So what is it?
Stationary points of inflexion
• This type of stationary point is called a
stationary point of inflexion.
First Derivative Test
• So, we have 3 outcomes from the first derivative test.
• If the sign of f’(x) changes from positive to negative
either side of the stationary point then it is a maximum
point.
• If the sign of f’(x) changes from negative to positive
either side of the stationary point then it is a minimum
point.
• If the sign of f’(x) is the same on either side of the
stationary point then it is a stationary point of
inflexion.
Concavity and the second derivative
• If the tangent to the curve is positioned below
the curve then the curve is called concave .
y’ < 0
y’ > 0
y’ = 0
• if y” > 0 then the curve is concave .
Concavity and the second derivative
• If the tangent to the curve is positioned above
the curve then the curve is called convex.
y’ = 0
y’ > 0
y’ < 0
• if y” < 0 then the curve is convex