Results 1 to 5 of 5

Math Help - Show that the cubic equation has one positive and two negative roots. How?

  1. #1
    Member ssadi's Avatar
    Joined
    Oct 2008
    Posts
    104

    Show that the cubic equation has one positive and two negative roots. How?

    Show that the equation x^3-12x-7.2=0 has one positive and two negative roots. Obtain the positive root to 3 significant figures using the Newton-Raphson process.
    How do i do the blue part. I am clueless.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by ssadi View Post
    Show that the equation x^3-12x-7.2=0 has one positive and two negative roots. Obtain the positive root to 3 significant figures using the Newton-Raphson process.
    How do i do the blue part. I am clueless.
    Let f(x) = x^3 - 12x - 7.2

    Note that f(x) is continuous for all values of x. Apply the Intermediate Value Theorem:

    f(0) < 0 and f(4) > 0 => f(a) = 0 for 0 < a < 4.

    f(-4) < 0 and f(-3) > 0 => f(b) = 0 for -4 < b < -3.

    f(-3) > 0 and f(-1/2) < 0 => f(c) = 0 for -3 < c < -1/2.

    Therefore two negative roots x = b and x = c and one positive root x = a.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Peritus's Avatar
    Joined
    Nov 2007
    Posts
    397
    let's look at the derivative of this function:

    f'(x) = 3x^2  - 12

    as you can see the function has to extrema points: a minima at (2,-23.2) and a maxima at x = (-2,8.8).

    We can also observe that:
    <br /> <br />
\begin{gathered}<br />
  \mathop {\lim }\limits_{x \to \infty } {\kern 1pt} \,f(x) = \infty  \hfill \\<br />
  \mathop {\lim }\limits_{x \to  - \infty } {\kern 1pt} \,f(x) =  - \infty  \hfill \\ <br />
\end{gathered} <br />

    and also that
    f(0) =  - 7.2

    Thus using the aforementioned facts and the fact that the function is continues one can easily conclude that f(x) has one negative root between 0 and -2, another negative root that's smaller than -2, and a positive root larger than 2.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member ssadi's Avatar
    Joined
    Oct 2008
    Posts
    104
    Quote Originally Posted by Peritus View Post
    let's look at the derivative of this function:

    f'(x) = 3x^2  - 12

    as you can see the function has to extrema points: a minima at (2,-23.2) and a maxima at x = (-2,8.8).

    We can also observe that:
    <br /> <br />
\begin{gathered}<br />
  \mathop {\lim }\limits_{x \to \infty } {\kern 1pt} \,f(x) = \infty  \hfill \\<br />
  \mathop {\lim }\limits_{x \to  - \infty } {\kern 1pt} \,f(x) =  - \infty  \hfill \\ <br />
\end{gathered} <br />

    and also that
    f(0) =  - 7.2

    Thus using the aforementioned facts and the fact that the function is continues one can easily conclude that f(x) has one negative root between 0 and -2, another negative root that's smaller than -2, and a positive root larger than 2.
    Less trial and error. I will use that method then.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by ssadi View Post
    Show that the equation x^3-12x-7.2=0 has one positive and two negative roots. Obtain the positive root to 3 significant figures using the Newton-Raphson process.
    How do i do the blue part. I am clueless.
    Descartes rule of signs shows this has exactly one positive real root.

    x^3-12x-7.2 is positive when x=-2, so there must in addition be two negative roots (since the cubic is continuous and it is negative at x=0 and again for large x).

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Quadratic and cubic equation -show that -(common roots)
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: August 19th 2011, 11:47 PM
  2. [SOLVED] +/- ve notation for roots, positive/negative, etc.
    Posted in the Algebra Forum
    Replies: 16
    Last Post: May 9th 2011, 10:57 AM
  3. Roots of a cubic equation
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: January 3rd 2010, 12:59 PM
  4. Roots of cubic equation
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: October 11th 2009, 05:35 AM
  5. Negative/positive Roots?
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 29th 2009, 11:29 AM

Search Tags


/mathhelpforum @mathhelpforum