Calculus and Geometry with a Cubic Function and Straight Line

[seniorhs9]

Homework Statement

I am asking about part iv).

[PLAIN]http://img715.imageshack.us/img715/7977/113ivb.jpg

Homework Equations

I guess they would be the ones in the earlier parts...

The Attempt at a Solution

In the given fact, I think $x^3 - x - m(x - a)$ distance from the cubic function to the line. So for every point in $(b, c)$, this would be negative.

I'm really not sure how to show $c = -2b$ so I just tried to play with some algebra...

At x = b... $b^3 - b - m(b - a) = 0$

At x = c... $c^3 - c - m(c - a) = 0$

So they're both equal to 0...


$b^3 - b - m(b - a) = c^3 - c - m(c - a)$

so $b^3 - b - mb = c^3 - c - mc$

so by part i) $b^3 - b - b(3b^2 - 1) = c^3 - c - c(3b^2 - 1)$

so $b^3 - b - 3b^3 + b = c^3 - c - 3b^2c + c$

so $-2b^3 = c^3 - 3b^2c$

so $b^2(3c - 2b) = c^3$

but this doesn't look useful...

Thank you.

 

[hotvette]

Have you answered i, ii, and iii? Re iv, it looks like you didn't use the relation given in the problem (the one you don't have to prove).

 

[seniorhs9]

=hotvette;3598514]Have you answered i, ii, and iii? ).

Yes..

Re iv, it looks like you didn't use the relation given in the problem (the one you don't have to prove).

Really? I think I did use it...

I got $b^3 - b - m(b - a) = 0$ by putting $x = b$ in the given relation...

 

[hotvette]

You might try doing the algebra first before substituting.

 

[seniorhs9]

You might try doing the algebra first before substituting.

I did try that and got up to $b^3 - b - mb = c^3 - c - mc$ which is what I wrote ...

Maybe you mean something else?

 

[hotvette]

Yep, I see what you mean. I spoke before trying it (gotta stop doing that, it burns me too often). One thing you can do (not sure it would be accepted by your prof though) is to substitute c=-2b into your last expression and see if both sides are indeed equal.

 

[seniorhs9]

Unfortunately, I don't think that's the best and proper way in math to do this? You'd have to work to show $c = -2b$ and not assume it's true already...

 

[hotvette]

Got it. Evaluate the given equation at x=a. After rearranging, you'll have an expression for c that is a ratio of polynomials in a & b. Substitute the expression for a and simplify.