cubics

**usagi_killer** · February 2nd 2010, 05:23 AM

Prove that for a cubic of the form $a(x-b)^3+d$ then there exists at least one set of real solutions $\{a,b,d\}$ such that:

$y_1=a(x_1-b)^3+d$

$y_2=a(x_2-b)^3+d$

$y'_1 = 3a(x_1-b)^2$

**HallsofIvy** · February 2nd 2010, 09:20 AM

Originally Posted by usagi_killer

Prove that for a cubic of the form $a(x-b)^3+d$ then there exists at least one set of real solutions $\{a,b,d\}$ such that:

$y_1=a(x_1-b)^3+d$

$y_2=a(x_2-b)^3+d$

$y'_1 = 3a(x_1-b)^2$

Strictly speaking, what you want to prove here is NOT true! If $x_1= x_2$ and $y_1\ne y_2$ , the first two conditions are incompatible. I am going to assume that $x_1\ne\x_2$ .

Since $y'= 3a(x- b)^2$ that third condition follows from the first so we really have only two conditions. That means we are free to choose one of the parameters to be whatever we want. Choosing $b= x_1$ makes the first condition $y_1= d$ . With $b= x_1$ and $d= y_1$ , the second condition becomes $y_2= a(x_2- x_1)^3+ y_1$ which gives $a= \frac{y_2-y_1}{(x_2- x_1)^3}$ as long as $x_1\ne x_2$ .

If we have $x_1= x_2$ and $y_1= y_2$ , then we really only have one condition. Choose $b= x_1$ and $d= y_1$ . a now can be any number at all.

**usagi_killer** · February 2nd 2010, 07:06 PM

Sorry $x_1 \neq x_2$ and $y_1 \neq y_2$

**Pulock2009** · February 2nd 2010, 08:25 PM

hey this comes under theory of equations,right??? i expanded (x-b)^3 and got a cubic. if the roots be y1,y2,y3 then i got the following:
y1+y2+y3=-3b
y1y2+y2y3+y3y1=-3b^2
y1y2y3=d/a-b^3

as evident there should be 3 roots but i simply couldnot find y1,y2,y3.there seems to be some kind of a proof not by usual methods.

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