I am asking about part iv).
I guess they would be the ones in the earlier parts...
The Attempt at a Solution
In the given fact, I think [itex] x^3 - x - m(x - a) [/itex] distance from the cubic function to the line. So for every point in [itex] (b, c) [/itex], this would be negative.
I'm really not sure how to show [itex] c = -2b [/itex] so I just tried to play with some algebra...
At x = b... [itex] b^3 - b - m(b - a) = 0 [/itex]
At x = c... [itex] c^3 - c - m(c - a) = 0 [/itex]
So they're both equal to 0...
[itex] b^3 - b - m(b - a) = c^3 - c - m(c - a) [/itex]
so [itex] b^3 - b - mb = c^3 - c - mc [/itex]
so by part i) [itex] b^3 - b - b(3b^2 - 1) = c^3 - c - c(3b^2 - 1) [/itex]
so [itex] b^3 - b - 3b^3 + b = c^3 - c - 3b^2c + c[/itex]
so [itex] -2b^3 = c^3 - 3b^2c [/itex]
so [itex] b^2(3c - 2b) = c^3 [/itex]
but this doesn't look useful...
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