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Conditions for Real-valued Roots


Date: 04/01/98 at 13:32:19
From: Heather
Subject: Calculus

Given the equation ax^3 + bx^2 + c = 0, use the first and second 
derivatives to help find conditions that ensure the equation has 3 
distinct real-valued roots.

So far I've figured this out....

   f'(x) = 3ax^2 + 2bx
         = x(3ax + 2b)
         = 3ax + 2b = 0
     3ax = -2b

           -2b
       x = ---- or 0
            3a
            

  f''(x) = 6ax + 2b = 0
     6ax = -2b         

            -b  
       x = ---
            3a

Also, a does not equal 0, and b does not equal 0; and f'(x) is a 
parabola and f''(x) is a line.

So where do I go from here?


Date: 04/01/98 at 16:41:41
From: Doctor Rob
Subject: Re: Calculus

f(x) has three real roots if and only if

   g(x) = f(x)/a,
        = x^3 + (b/a)*x^2 + c/a,
        = x^3 + B*x^2 + C,

has three real roots. We know that for large enough positive x, 
g(x) > 0, and for small enough negative x, g(x) < 0.

You have found the two critical points x = 0, y = C, and x = -2*B/3,
y = (4*B^3 + 27*C)/27. Call these points P and Q, respectively. At 
these points, g''(0) = 2*B, and g''(-2*b/3) = -2*B. Thus one is a 
local maximum, the other a local minimum, by the appropriate second 
derivative test. A root means that the curve y = g(x) crosses the 
axis, that is, changes sign. Moving from large values of x where
g(x) > 0, to smaller values, we must cross the axis, reach a local 
minimum, cross the axis again, reach a local maximum, then cross the 
axis a third time, before reaching the very small values of x for 
which g(x) < 0 is guaranteed. There are two cases, depending on the 
sign of B.

  Case 1: B > 0. Then P is the local minimum and Q is the local 
          maximum. Then there must be a zero between P and +infinity,                    
          one between P and Q, and one between Q and -infinity. That
          means that C < 0 and (4*B^3+27*C)/27 > 0. The last condition
          is equivalent to 4*B^3 + 27*C > 0, or C > -4*B^3/27.                     
          Thus B > 0 > C > -4*B^3/27.

  Case 2: B < 0. Then Q is the local minimum and P is the local 
          maximum. Then there must be a zero between Q and +infinity,
          one between P and Q, and one between P and -infinity. That                                 
          means that C > 0 and (4*B^3+27*C)/27 < 0. The last condition
          is equivalent to 4*B^3 + 27*C < 0, or C < -4*B^3/27.
          Thus B < 0 < C < -4*B^3/27.

In either case, B and C must have opposite signs, and B and 
4*B^3 + 27*C must have the same sign.

In terms of a, b, and c,

   b > 0 > c > -4*b^3/27*a^2, or
   b < 0 < c < -4*b^3/27*a^2.

-Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   


Date: 04/01/98 at 16:56:24
From: Doctor Anthony
Subject: Re: Calculus

The method is as follows.

First find conditions for the quadratic f'(x) to have real, distinct 
roots.

   f'(x) = 3ax^2 + 2bx = 0    so x = 0 and  x = -2b/3a

Now you require the two turning points to be on opposite sides of the 
x axis, so (assuming a and b have same sign) we require

   f(-2b/3a) > 0  and f(0) < 0    so c < 0    

If a and b have opposite signs, we require c > 0.

   a(-2b/3a)^3 + b(-2b/3a)^2 + c > 0

   -8b^3/(27a^2) + 4b^3/(9a^2) + c > 0

   (4/27)(b^3/a^2) + c > 0

   4b^3 + 27a^2*c > 0

If a and b have the same sign, then c < 0 and we require  b > 0 and 

    |4b^3| > 27a^2*c

If a and b have opposite signs, then c > 0  and

    4b^3 + 27a^2*c < 0

So we require b < 0 and |4b^3| > 27a^2*c

We can summarize as follows:

   if a and b same signs then c < 0 and |4b^3| > 27a^2*c

   if a and b opposite signs then b < 0, C > 0, and  
                                |4b^3| > 27a^2*c

These would be sufficient conditions for three real roots. There is no 
need to use the second derivative as the above are sufficient. There 
is a point of inflection, given by f''(x) = 0, and this occurs at 
x = -b/(3a), i.e. halfway between the maximum and minimum points.

-Doctor Anthony, The Math Forum
Check out out web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus

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