Stationary Points
A stationary point of a function is a value in the domain at which . At these points the function could possibly have a
maximum value or a minimum value, but it's not guaranteed. You have to check to see whether or not the sign of the derivative changes
on either side of the point.
One way to determine if you have a max or min is to use the First Derivative Test. If the derivative changes from positive to negative
at a stationary point, then you have a max. If the derivative changes from negative to positive at a stationary point you have a min.
Blue Question: (CEMC vol.8) Consider the function
where
is a positive constant. Prove that .
STEP 1999, Math II, #7: The curve C has equation
where the square root is positive. Show that, if
, then C has exactly one stationary point. [It's useful to consider what the
domain of the function is for various values of
. Then consider under what circumstances there is a stationary point in the domain.
A sketch of the graph indicates there are three distinct cases to consider.]
Sketch C when (i)
and (ii) .
STEP 2004, Math II, #3: The curve C has equation
Show that the gradient of C is and find the coordinates of all the stationary points. Determine the nature of
each stationary point and sketch C.
In separate diagrams draw sketches of the curves whose equations are:
(i)
(ii)
In each case, you should pay particular attention to the points where the curve meets the
axis.
STEP 2002, Math I, #2: Let
, where and are both integers greater than 1. Show that the curve
has a stationary point with
. By considering show that this stationary point is a maximum if is even and a
minimum if
is odd.
Sketch the graphs of
in the four cases that arise according to the values of and . [Hint: You can use either the First Derivative
Test for extrema or the Second.]
STEP 2008, Math II, #3: (i) Find the coordinates of the turning points of the curve . Sketch the curve and
deduce that
for all .
Given that each of the numbers
, , and lies between 0 and 1, prove by contradiction that at least one of the numbers ,
and is less than or equal to .
(ii) Given that each of the numbers
and lies between 0 and 1, prove that at least one of the numbers and is less
than or equal to
.
STEP 2012, Math I, #1: The line L has equation
, with and . It passes through the point and cuts
the axis at the points
and , where , , and are all positive. Find and in terms of , and .
As L varies with
remaining fixed, show that the minimum value of the sum of the distances of and from the origin is
and find a similar form for the minimum distance between
and . (You may assume that any stationary values of these distances are minima.)
STEP 1998, Math III, #1: Let
for
. Sketch the curve , giving the coordinates of the stationary points.
Now let
, , .
Show that the stationary points of occur at the same values of as those of , and find the corresponding values
of
.
Explain why, if or , cannot be arbitrarily large.
STEP 2003, Math III, #5: Find the coordinates of the turning point on the curve . Sketch the curve in the case that
the equation
has two distinct real roots. Use your sketch to determine necessary and sufficient conditions on and
for the equation
to have two distinct real roots. Determine necessary and sufficient conditions on and for this
equation to have two distinct positive roots.
Find the coordinates of the turning points on the curve (with ) and hence determine necessary and sufficient
conditions on
and for the equation to have three distinct real roots. Determine necessary and sufficient conditions
on
, and for the equation to have three distinct positive roots.
Show that the equation has three distinct positive roots.
STEP 2012, Math II, #5: (i) Sketch the curve , where
,
and
is a constant.
(ii) The function is defined by
,
where
and are constants, and . Sketch the curve in the two cases and , finding the values
of
at the stationary points.