Question

Find the stationary points on the curve y = x^3 + 6x^2 + 9x − 11 and determine their nature.

Answer 1

This is what I've done:

y' = 2x + 12x + 9
y' = 0
Therefore 2x + 12x + 9 = 0...

But it seems wrong.

Answer 2

Make sure you calculate your derivative properly. If\[y = x^3 + 6x^2 +9x-11\]
then \[y' = 3x^2 + 12x + 9\]

Answer 3

Ah, I see. So x = -3 or -1?

Answer 4

Yes

Answer 5

Really x= -3 and x = -1 are both solutions to the equation.

Answer 6

So how do I work out the stationary points and nature from here?

Answer 7

Well since the derivative of y is zero at those points, those are your stationary points. You may have to do some testing around those points to determine whether or not each point is a maximum, a minimum, or an inflection point.

Answer 8

Do I use a table of values or the second derivative...?

Answer 9

Yes you would want to do that to determine concavity. You would use a table of value for the first derivative to determine intervals of increasing and decreasing function values.

Answer 10

I draw out a sort of number line for these problems; it's easier for me to see what intervals y is increasing and decreasing that way.