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Stationary and Turning Points

Stationary and turning points are points at which the curve changes its direction (turns around). Stationary points are also called turning points.

There are 3 types of stationary points:

  • Minimum point
  • Maximum point
  • Point of horizontal inflection

We call the turning point (or stationary point) in a domain (interval) a local minimum point or local maximum point depending on how the curve moves before and after it meets the stationary point.

 

The curve here decreases on the left of the stationary point and increases on the right. It decreases and reaches a minimum point and then increases. This is a local minimum point.

local minimum point
x LHS Minimum point RHS
f'(x) f'(x)<0 f'(x)=0 f'(x)>0

 

The curve here increases on the left of the stationary point and decreases on the right. It increases and reaches a maximum point and then decreases. ThisĀ is a local maximum point.

local maximum point
x LHS Minimum point RHS
f'(x) f'(x)>0 f'(x)=0 f'(x)<0

 

The local maximum point and the local minimum point are also called relative maxima and relative minima respectively because the maximum or minimum point reached is relative to the interval where it occurs.

 

A point of horizontal inflection occurs in a monotonic increasing or monotonic decreasing curve. The curve is either increasing or decreasing on both sides of this point. The curve does not turn and hence it is not a turning point, and the curve is stationary at this point.

The derivative of the curve has the same sign on both sides of the inflection point.

x LHS Minimum point RHS
f'(x) f'(x)>0f'(x)<0 f'(x)=0 f'(x)>0f'(x)<0

 

If we restrict the domain of the curve then the curve will have an absolute maximum and minimum value. The absolute maximum is the greatest value of the curve in the domain and the absolute minimum is the least value of the curve in the domain.

 

Now let us look at a few examples of stationary points.