More Differentiation: Stationary Points

You need to be able to find a stationary point on a curve and decide whether it is a turning point (maximum or minimum) or a point of inflexion.

Finding the Stationary Point:

Looking at the 3 diagrams above you should be able to see that at each of the points shown the gradient is 0 (i.e. the curve goes flat). In lesson 6 we said that differentiationgives us the gradient function. So to find any stationary points we need to differentiatefirst then put the resulting gradient function equal to zero.

When (i.e. the gradient is zero) you have a stationary point.

In short:find the 1st differential,put it equal to zero and solve.

Examples:

Find the stationary point on the curve :

Step 1: Differentiate to give:

Step 2: Put this equal to zero:

Step 3: Solve:

We now know that the stationary point is when.

If we require the full co-ordinate we need to find the value of when .

(We are looking for so we need to use the original equation, )

Original equation:

Substitute:

Solve:

So the stationary point is at

Find the stationary point on the curve :

Step 1: Differentiate to give:

Step 2: Put this equal to zero:

Step 3: Solve:

We now know that there are 2 stationary points: one at and one at .

Next find the corresponding values,using the equation,

When :

So the 2stationary pointsare at and

In the above examples we have found out where the stationary points are but we don’t know what type they are (i.e. maximum, minimum or point of inflexion). We now need to go a step further (and differentiate again) to find the type.

Type of Stationary point:

To find the type of stationary point we need to differentiate again (i.e. find the 2nd differential, denoted by ). This measures the change in gradient and can help to decide whether a stationary point is a maximum, minimum or point of inflexion.

After finding where the stationary point is, find the 2nd differential and if:

> 0 then you have a minimum (positive is minimum).

< 0 then you have a maximum (negative is maximum).

= 0 then it could be maximum, minimum or point of inflexion.

In the last case – differentiate again:

If 0 then it is a point of inflexion.

Remember:1st differential tells us where the stationary point is.

2nd differential tells us what type the stationary point is.

Examples:

Find the stationary point on the curve :

Use 1st differential to find where the Stationary point is:

Step 1: Differentiate to give:

Step 2: Put this equal to zero:

Step 3: Solve:

Find corresponding value:

Substitute :

Solve:

So the stationary point is at

Use 2nddifferential to find what type the Stationary point is:

= 2This is positive (>0) so the turning point must be a minimum.

Find the stationary points on the curve :

Use 1st differential to find where the Stationary point is:

Step 1: Differentiate to give:

Step 2: Put this equal to zero:

Step 3: Solve:

Find corresponding values:

Substitute :

So the stationary pointsare at and (2, 0)

Use 2nddifferential to find what type the Stationary points are:

This looks a bit different from the 1st example as there is an in the 2nd differential. We now need to substitute each value of in, separately, to find out what type of stationary point each one is.

When :

(maximum)

When :

(minimum)

So (0, 4) is a maximum and (2, 0) is a minimum.

Find the stationary point on the curve :

Step 1: Differentiate to give:

Step 2: Put this equal to zero:

Step 3: Solve:

Find corresponding value:

Substitute :

Solve:

So the stationary point is at

Use 2nddifferential to find what type the Stationary point is:

When = 0: so could be a max, min or point of inflexion.

Use 3rddifferential to decide:

The 3rd differential is not 0 so it is a point of inflexion.