How can I determine the stationary points of a curve and their nature?

For example, y = 3x³+ 9x² + 2. Determine the stationary points and their nature.

Let's remind ourselves what a stationary point is, and what is meant by the nature of the points.

A stationary point is a point on a curve where the gradient equals 0.

The nature of a stationary point is:

A minimum - if the stationary point(s) substituded into d²y/dx² > 0

A point of inflection - if the stationary point(s) substituded into d²y/dx² = 0 and d²y/dx² of each side of the point has different signs.

A maximum - if the stationary point(s) substituded into d²y/dx² < 0

Back to our question...

y = 3x³+ 9x² + 2. Determine the stationary points and their nature of the curve.

In order to determine the stationary points, we need to differentiate y to get dy/dx. Using standard differentiation...

dy/dx = 9x² + 18x

We now need to equate dy/dx = 0, as dy/dx = 0 at stationary points.

⇒ 0 = 9x² + 18x

Solving for x by factorising, we get

⇒ 0 = 3x(3x + 6)

so x = 0 or x = -2.

We have the x values of the stationary points, now we can find the corresponding y values of the points by substituing the x values into the equation for y.

For x = 0,

y = 3(0)³ + 9(0)² + 2 = 2

So (0, 2) is a stationary point.

For x = -2

y = 3(-2)³ + 9(-2)² + 2 = 14

So (-2, 14) is a stationary point.

In order to determine the nature of the points, we need to work out d²y/dx². This means we need to differentiate our dy/dx. By using standard differentiation...

dy/dx = 9x² + 18x

⇒ d²y/dx² = 18x + 18

We will now substitute both of our stationary points into d²y/dx²to determine their nature.

For x = 0

d²y/dx² = 18

As 18 > 0, this point is a minimum.

For x = -2,

d²y/dx² = -18.

As -18 < 0, this point is a maximum.

Our answer is: Stationary point 1 is (0, 2) - a minimum, and stationary point 2 is (-2, 14), a maximum.

Here are a few more questions to test your understanding, scroll down for the answers!

Find the stationary points on the curve y = 3x² - 12x, and determine their nature.
Find the stationary points on the curve y = x²e^x, and determine their nature.

Answers