Stationary Points

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

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:

. 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

, , and lies between 0 and 1, prove by contradiction that at least one of the numbers ,

and lies between 0 and 1, prove that at least one of the numbers and is less

than or equal to

, with and . It passes through the point and cuts

the axis at the points

remaining fixed, show that the minimum value of the sum of the distances of and from the origin is

and . (You may assume that any stationary values of these distances are minima.)

STEP 1998, Math III, #1: Let

. 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.

, 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

and are constants, and . Sketch the curve in the two cases and , finding the values

of

at the stationary points.