C2 Differentiation!!!

The equation of a curve is (x+1)(x-3)³

i) Write the equation of the curve in the form y=ax⁴+bx³+cx²+dx+e
ii) Find the co-ordinates of the points where dy/dx = 0

iii) Classify the stationary points
iv) Sketch the curve

I have question i as y= x⁴ - 8x³ +18x² - 27
and the I have got (3,0) for one of the coordinates but not sure if i've done it right

Yeah that's right so far. There is one more stationary point.

The only other i got was (0,-27) but didn't think this was correct

It is. Why do you think it wasn't?

i don't know..... whats my next step

You should be aware of what you are doing and why you are doing it when performing Maths at this level - so being uncertain about a correct answer and WHY it is correct is something you should look into.

Well "classify" refers to the determination of the nature of the stationary points. How do you find out whether a stationary point is a maximum or a minimum?

Okay so i think i've got (3,0) as the minimum but i don't get a maximum when i use (0, -27)

Okay so i think i've got (3,0) as the minimum but i don't get a maximum when i use (0, -27)

Can you please post all the working you've done so far to find and clasify the stationary points?

Your working shows that (0,-27) is a minimum since the second derivarive is > 0. But you said it was a maximum in your last post.

For the other stationary point you got $\frac{d^2 y}{dx^2} = 0$. This is neither > 0 nor < 0 so you still can't clasify the stationary point. Do you know what to do in a case like this?

As x=3 gives 0 in the second derivative, it means it is a stationary point of inflection - there is no necessary maximum point I believe.

So as the second derivative is equal to 0 its either a point of infelction or a stationary point?

Point of inflection : A point on a graph where there is a change in the direction of curvature (may not be a stationary point)

Stationary point : A point where the gradient is 0

Stationary point of inflection : A point where the gradient is 0 and there is a change in the direction of curvature

If the first derivative is 0 then this means you have a stationary point, which could be maximum, minimum or a stationary point of inflection. If the second derivative is equal to 0 as well as the first derivative then you have a stationary point but you still don't know what type.

In this situation, it's best to consider the gradient either side of the stationary point. Have you done this before? (There is another method which involves checking higher order derivatives but this often not taught at C2).

I have yes, if its the first derivative that you use when x is 2.9 i got 0.116 and then when x is 3.1 i got 0.124, so what does this mean?

Have a look at this diagram:



The bottom two images show that if the gradients on either side of the stationary point are both negative or both positive then the stationary point is a point of inflection.

So would you call it a positive point of inflection?

No "a point of inflection" is all you need to say.