A Level Maths Notes: C3 - Curves and Maxima/Minima/Stationary Points/Turning Points
I don't know why these thing need four possible names. We can call them stationary points and classify them as maxima or minima.
Definition: A stationary point is a point on a curve
where
Definition: A stationary point, with
is
a Minima if
On
a graph a minimum is lower than the points on either side as on the
graph below.
Definition: A stationary point, withis
a Maxima if
A
maximum is above the points on either side of it.
We often have to find the stationary points and classify them as maxima or minima. To do this,
We find
and
solve
to
find the x – coordinates of the turning points.
We put these values of x into the equation of the graph to find the y – coordinates of the tuning points.
We differentiate
to
find
We
put the values of x from 1. into the expression for
to
find a value: if this value is positive then the stationary point is
a minimum. If this value is negative then this stationary point is a
maximum. If this value is zero then we can find y values a little
either side of the stationary point and do a 3 point sketch to
determine if a maximum, minimum.
Example:
Find the turning points and determine their nature.
This
is shown below.
