Stationary points are points on a curve where the function is neither increasing or decreasing. That is, f(x) is not changing at the stationary point. We call it stationary because when we envision an object moving along the curve f(x), the object stops there. The object will be stopping at that point.
A stationary point on a function f(x) is a point where the derivative of f(x) is equal to 0. At the stationary point, the tangent to a curve is a line parallel to the horizontal axis. It has a zero gradient.
The graph below represents the curve x2-6x+5
If a tangent is drawn at point (3,4), it would be a horizontal line parallel to the x -axis.
To the left of the point (3,4), the gradient of the curve is negative and to the right the gradient is positive.
The general derivative of the curve is given as:
and the derivative dy/dx at x=3 is given as 2(3)-6 = 0
A point where gradient of a curve is zero is known as a stationary point.
A point where gradient of the curve changes from negative to positive is referred to as the minimum point. It is the lowest point on a curve.
A point where gradient of curve changes from positive through zero to negative is called a maximum point.
The maximum and minimum points are called the turning points since at those points the curves changes direction
sometimes the curve does not change direction but flatten to zero before continuing in the same direction of positive or negative orientation.
A point where gradient of a curve moves from positive through zero and to positive again or from negative through zero to negative again is known as an inflection point.
Here is a summary of types of stationary points and how to identify them.
Example question
Identify the stationary points on the curve y = 2x3-6x +5 and determine nature of each of the stationary point.
solution
We start by defining the gradient of our curve which is the derivative of y with respect to x. That is ;
at stationary point:
hence 6x2 – 6 = 0
Factoring out 6, we get:
solving for x we get:
x =-1 or x = 1
when x = -1, y= 2(-1)3-6(-1)+5 = -2 + 6 +5 =9
when x= 1, y = 2(1)3-6(1)+5 = 2 -6 +5 = 1
hence the stationary points are (-1, 9) and (1, 1)
at x = -1, consider a point on its immediate left, say -2.
the gradient at -2 = 6(-22) -6 = 6(4)-6 = 18 which is positive
consider a point on the right of -1 such as x=0
the gradient of the curve becomes, 6(02)-6 = -6 which is a negative gradient. Here is the representation of gradient change from the left to right of the point (-1, 9).
Therefore, point (-1,9) is a maximum point.
consider the point (1, 1)
on the left of the point say x= 0, the gradient =6(02)-6 = -6 which is a negative gradient
To the right of (1,1), say x = 2, the gradient = 6(22)-6= 18 which is positive. The change of gradient from left of (1, 1) through x=1 to the right is as shown.
Therefore point (1, 1) is a minimum point.
Examination Practice Questions on stationary points
The equation of a curve is given as y = (x -1)(x2 +x-2)
a) Find i) The y-intercept of the curve. (1 mark)
ii) The x- intercepts of the curve. (2marks)
b) i) Find the stationary points of the curve. (3marks)
ii) Determine the nature of the stationary points. (2marks)
c) Hence sketch the curve. (2mks)
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