Maths help.

Iwasfrozen

Hey can anyone help me out with the LC question ?

Find the range of values of c for which f(x) = 0 has three distinctreal roots.

f(x) = 2x^3 + 3x^2 - 12x + c = 0

Local max at x = -2

Call Me Jimmy

do ye not have yer maths book or somethin?

pjmn

Think this should help...

http://en.wikipedia.org/wiki/Cubic_function

Best of luck.

Iwasfrozen

Call Me Jimmy wrote: »

do ye not have yer maths book or somethin?

Yes I do. Have you not got your grammer book or something ?

Call Me Jimmy

how original, it shows you how to do this exact thing in your maths book and you ask people to help you with no sign of effort.

Iwasfrozen

Call Me Jimmy wrote: »

how original, it shows you how to do this exact thing in your maths book and you ask people to help you with no sign of effort.

I have tried several ways.

I can't put a number in for x because of the c and I can't use (b+or-(b^2-4ac)^1/2)/2a because it's a cubic.

MathsManiac

Give people a chance to help you, for goodness sake, by giving them the whole question at least! The question can't be answered as it stands.

This is a part (ii) of an LC question. You can't answer it until you've first of all found the value of b, (which was part (i),) and to do that you need to use the extra piece of information that you didn't bother to give. (i.e. the fact that there's a local max at x=-2).

The solution is fully done out in the marking scheme on the exams commission's website (paper 1, Q6(c), on P26):
http://www.examinations.ie/archive/markingschemes/2008/LC003ALP000EV.pdf

D-Generate

MathsManiac wrote: »

Give people a chance to help you, for goodness sake, by giving them the whole question at least! The question can't be answered as it stands.

This is a part (ii) of an LC question. You can't answer it until you've first of all found the value of b, (which was part (i),) and to do that you need to use the extra piece of information that you didn't bother to give. (i.e. the fact that there's a local max at x=-2).

The solution is fully done out in the marking scheme on the exams commission's website (paper 1, Q6(c), on P26):
http://www.examinations.ie/archive/markingschemes/2008/LC003ALP000EV.pdf

Yeah! What he said!

Iwasfrozen

MathsManiac wrote: »

Give people a chance to help you, for goodness sake, by giving them the whole question at least! The question can't be answered as it stands.

This is a part (ii) of an LC question. You can't answer it until you've first of all found the value of b, (which was part (i),) and to do that you need to use the extra piece of information that you didn't bother to give. (i.e. the fact that there's a local max at x=-2).

The solution is fully done out in the marking scheme on the exams commission's website (paper 1, Q6(c), on P26):
http://www.examinations.ie/archive/markingschemes/2008/LC003ALP000EV.pdf

sorry, b = -12 And there is a local max at x = -2

Iwasfrozen

MathsManiac wrote: »

The solution is fully done out in the marking scheme on the exams commission's website (paper 1, Q6(c), on P26):
http://www.examinations.ie/archive/markingschemes/2008/LC003ALP000EV.pdf

Also I don't want to look at the marking scheme as I will never improve that way. If someone told how to start I could probably do the rest myself.

Fremen

Call me jimmy, you sound like you're plodding toward a modding...

Iwasfrozen:
Ask yourself
"What am I told, and how can I use it?".
This is the first thing you should ask every time you start a problem.

You have some function f(x). You know there's a local max at x=-2. So how is a local max characterised in terms of derivatives?

Iwasfrozen

Fremen wrote: »

Call me jimmy, you sound like you're plodding toward a modding...

A modding ? wait, what, why ?

Fremen

Eh, he's more or less trolling and not being helpful. Where's lexlipred when you need him?
I edited my older comment, hopefully it's enough to get you started.

Iwasfrozen

Fremen wrote: »

Call me jimmy, you sound like you're plodding toward a modding...

Iwasfrozen:
Ask yourself
"What am I told, and how can I use it?".
This is the first thing you should ask every time you start a problem.

You have some function f(x). You know there's a local max at x=-2. So how is a local max characterised in terms of derivatives?

At a max:
f(x) = 0
f'(x)<0

MathsManiac

Iwasfrozen wrote: »

Also I don't want to look at the marking scheme as I will never improve that way. If someone told how to start I could probably do the rest myself.

Having found b, find the local max and the min in terms of c. Then ask yourself: by looking at the graph of a cubic, how would you know whether it has three real roots or not, and what does that tell you about the max and min points?

Iwasfrozen

MathsManiac wrote: »

Having found b, find the local max and the min in terms of c. Then ask yourself: by looking at the graph of a cubic, how would you know whether it has three real roots or not, and what does that tell you about the max and min points?

I can't find the max in terms of c, the c disappears when it is differentiated.

Fremen

Hm, sorry, I didn't actually work out the answer before I responded. Usually with leaving cert questions you're given the specific information you need to solve the problem in the phrasing of the question, and the way you phrased the question threw me a bit. There's a local max at x=-2 for ALL values of c, rather than the ones for which the curve has real roots.

So you have a cubic. Same principle applies: what can you do with the information you're given? You want to know something about roots. You (should) know that roots and factors are intimately connected. What can you say about the factors if the roots are real?

Iwasfrozen

Fremen wrote: »

Hm, sorry, I didn't actually work out the answer before I responded. Usually with leaving cert questions you're given the specific information you need to solve the problem in the phrasing of the question, and the way you phrased the question threw me a bit. There's a local max at x=-2 for ALL values of c, rather than the ones for which the curve has real roots.

So you have a cubic. Same principle applies: what can you do with the information you're given? You want to know something about roots. You (should) know that roots and factors are intimately connected. What can you say about the factors if the roots are real?

If x = r, x = t and x = u are roots then x - r = 0, x - t = 0 and x - u = 0 are factors.

And (x - r ).(x - t).(x - u) = x^3 + ax^2 + bx + c

But I don't see how this helps.

Fremen

Well,
c = -tuv, for example. I was thinking you could equate the coefficients and work it out that way.

Sorry, I've more or less been talking out my arse, but I took a look at the marking scheme there. Apparently my approach is a "blunder". Best pay attention to mathsmaniac

MathsManiac

Iwasfrozen wrote: »

I can't find the max in terms of c, the c disappears when it is differentiated.

There's no c in the x-values of the max and min, but there is a c in the y-value. (Remember that the y-value of the point is found by subbing into the original function.)

The next bit might give the game away too much for you, so I'll use spoiler tags:

For the function to have three roots, the local max has to be above the x-axis and the local min has to be below it.

MathsManiac

Fremen wrote: »

Well,
Sorry, I've more or less been talking out my arse, but I took a look at the marking scheme there. Apparently my approach is a "blunder". Best pay attention to mathsmaniac

Can't win 'em all. And it's pretty rare for you to give anyone a bum steer around here!

Anyway, in my book, with any interesting problem, all plausible lines of attack are a good thing, as a matter of principle!

Iwasfrozen

MathsManiac wrote: »

There's no c in the x-values of the max and min, but there is a c in the y-value. (Remember that the y-value of the point is found by subbing into the original function.)

The next bit might give the game away too much for you, so I'll use spoiler tags:

For the function to have three roots, the local max has to be above the x-axis and the local min has to be below it.

Does this mean the local min = - 2 ? If so it the curve must pass the x-axis.

At the x axis y = 0: 2x^3 + 3x^2 - 12x + c = 0
dy/dx = 6x^2 + 6x - 12 = 0
d^2y/dx^2 = 12x + 6 > 0 local min = -2
d^2y/dx^2 = 24 + 6 > 0 True.

So this is true. But I don't know what to do next as the c disappears when I derive.

Unless I should put x = + or - 2 in for y ?

So that would give me:
y = 2(2)^3 + 3(2)^2 - 12(2) + c = 0
y = 16 + 12 - 24 + c = 0
y = c = -4

And:
y = 2(-2)^3 + 3(-2)^2 -12(-2) + c = 0
y = -16 + 12 + 24 + c = 0
y = c = -20

-4 > c > -20: Is this correct ?

MathsManiac

Iwasfrozen wrote: »

Does this mean the local min = - 2 ?

No. Surely you know how to find the max and min points of a cubic. You set dy/dx = 0 and solve. This gives you x=1 and x=-2. You know x=-2 gives the max, so x=1 gives the min. Now you get the two y-values:

f(-2) = 2(-2)^3 + 3(-2)^2 -12(-2) + c
= c+20
So the max is (-2, c+20)

f(1) = 2(1)^3 + 3(1)^2 -12(1) + c
=c-7
So the min is (1, c-7)

Now view the spoiler in my last post.

If you can't wrap it up from there, you really should just look at the marking scheme.

bnt

Best bit of advice I ever received for this kind of question: always draw the function. In this case, you can draw the function with c=0 to get an idea of what's going on, and an idea of how the answer should look:



The method for finding the local minima/maxima has already been discussed, and the c value simply serves to move the plotted function up or down on the plane. So, what value of c would put that local minimum on the x-axis? And the local maximum?