Stationary Points of Curves

stationary

From previous calculations, we understand that at any particular point, the gradient of a curve is positive when it is moving up at that point, the gradient is positive, and the gradient is negative when the curve is moving down. However, we call the curve stationary when a curve passes through a horizontal movement (gradient = 0). In other cases, inflexion points also occur.

When y' > 0, the curve is increasing

When y' < 0, the curve is decreasing

When y' = 0, the curve is stationary.

Examples

 

1. Find all x values for which the curve f(x)={x}^{2}-4x+2 is increasing.

Show Answer

f'(x)=2x-4 \\ f'(x)>0 \\ i.e. 2x-4>0 \\ \quad \quad 2x>4 \\ \quad \quad x>2

So the curve is increasing for x > 2.

 

 

2. Find any stationary points on the curve y={x}^{3}-48x-7.

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y'=3{x}^{2}-48

For stationary points,

3{x}^{2}-48=0 \\ 3{x}^{2} = 48 \\ {x}^{2}=16 \\ x=\pm 4

When x=-4, y={4}^{3}-48(4)-7=-135

When x=-4, y={(-4)}^{3}-48(-4)-7=121

So the stationary points are (4, -135) and (-4, 121).

 

 

Types of Stationary Points

 

 

The three types of stationary points are:

1. Local Minimum

Local minimum is the lowest value between a specified range of a function. The reason we say local is because it is possible that the largest and the smallest value of the whole function is at the endpoint of a graph. As seen from the graph above, the curve is decreasing on the left and increasing on the right of the minimum turning point. That is,

 

2. Local Maximum

On the other hand, the local maximum's curve is increasing on the left and decreasing on the right of the maximum turning point. That is,

 

Local maximum and minimum points are also called turning points, or maxima or minima.

To check whether a point is a maxima or minima, analyse the region to the left or right of the turning point in question, and derive a conclusion of the gradient of the curve.

 

3. Point of Inflexion

Another type of stationary points (gradient = 0) is a point of inflexion, where the curve stops increasing (or decreasing) at one point and then continued its increasing (or decreasing) pathway. Therefore, if the curve is increasing to the left side of a turning point (x, y), the curve will continue increasing to the right of the turning point. The same pattern appears when the curve is decreasing.

 

Examples

 

1. Find the stationary point on the curve y={x}^{3} and determine what type it is.

Show Answer

\cfrac { dy }{ dx } =3{ x }^{ 2 }\\ \\ \cfrac { dy }{ dx } =0\qquad (stationary\quad point)\\ \\ 3{ x }^{ 2 }=0\qquad \longrightarrow \qquad x=0\\ \\ When\quad x=0,\quad y={ 0 }^{ 3 }=0

So the stationary point is (0, 0). To determine its type, check the curve on the left and right side of point (0, 0).

Since the curve is increasing on both sides, (0, 0) is a point of inflexion.

 

 

2. Find any stationary points on the curve f\left( x \right) =2{ x }^{ 3 }-15{ x }^{ 2 }+24{ x }-7 and distinguish them.

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f^{ ' }\left( x \right) =6{ x }^{ 2 }-30x+24\\ f^{ ' }\left( x \right) =0\qquad (stationary\quad points)\\ 6{ x }^{ 2 }-30x+24=0\\ { x }^{ 2 }-5x+4=0\\ (x-1)(x-4)=0\\ \therefore \qquad x=1\quad or\quad 4\\ \\ Substitute\quad back\quad to\quad function:\\ \\ f\left( 1 \right) =2{ (1) }^{ 3 }-15{ (1) }^{ 2 }+24(1)-7=4\\ \\ f\left( 4 \right) =2{ (4) }^{ 3 }-15{ (4) }^{ 2 }+24(4)-7=-23

So, (1, 4) and (4, -23) are stationary points.

Next, we check the regions to the left and right of the stationary points.

For (1, 4):

Therefore, point (1, 4) is a local maximum turning point.

For (4, -23):

Therefore, point (4, -23) is a local minimum turning point.

 

 

Knowledge about stationary points is an absolutely crucial part of Curve Sketching. Make sure you master these techniques to be able to sketch any curve of a function.