Results 1 to 8 of 8

Math Help - quadratic cubic

  1. #1
    Newbie
    Joined
    Oct 2005
    Posts
    12

    Please help with this quadratic cubic

    I need help with the following.......please :-(

    Show that if X2 > X1 then X2 ^3 + 2X2 + 7 > X1 ^3 + 2X1 + 7 for all X2, X1 belonging to R. Without using calculus, what does this tell us about the graph of f(x) = X^3 + 2X + 7?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Sep 2005
    Posts
    20
    can you show f(x2)-f(x1)>0?

    if you can show for every x2>x1, we can have f(x2)>f(x1)
    this means this function is monotone increasing.

    if still having problem please point out.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2005
    Posts
    12
    I still don't get this I need the full explanation.... sorry
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2005
    Posts
    20
    alright

    to show X2 ^3 + 2X2 + 7 > X1 ^3 + 2X1 + 7

    ( X2 ^3 + 2X2 + 7 ) - ( X1 ^3 + 2X1 + 7) <------- this is f(x2)-f(x1)

    (x2^3-x1^3) + (2x2-2x1) + (7-7)

    Here you have 3 terms, and x2>x1

    x2^3 must be greater than x1^3, so x2^3-x1^>0
    2x2 must be greater than 2x1, so 2x2-2x1>0
    7-7 must be zero.

    sum of two greater than 0 terms and one 0 term, must be greater than 0.

    then we proved X2 ^3 + 2X2 + 7 > X1 ^3 + 2X1 + 7
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Oct 2005
    Posts
    12
    Thanks it's like saying this really

    X2 ^3 + 2X2 > X1 ^3 + 2X1 therefore X2 > X1 but better.

    Any help in the following part of the question?

    Without using calculus, using what we have just explained, what does this tell us about the graph of f(x) = X^3 + 2X + 7?

    About the roots, does this prove there is a point of inflextion, normally a cubic has 3 roots here there is only one etc etc...
    Last edited by alexis; October 22nd 2005 at 03:21 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    " Without using calculus, using what we have just explained, what does this tell us about the graph of f(x) = X^3 + 2X + 7? "

    What was explained:
    If (x2)^3 +2(x2) +7 > (x1)^3 +2(x1) +7,
    then x2 > x1.

    What is that to the graph of f(x) = x^3 +2x +7 ?

    Answer:
    >>>That means there is no horizontal tangent line to any point on the graph. Not even at the inflection point where x=0 and y=7, or point (0,7).

    >>>That means, at the positive half of the graph, which is to the right of the y-axis, the graph goes up as x approaches infinity.
    Example, if x1 = 0.1, x2 = 0.3, x3 = 0.5, .....
    f(0.1) = (0.1)^3 +2(0.1) +7 = 7.201
    f(0.3) = (0.3)^3 +2(0.3) +7 = 7.627
    f(0.5) = (0.5)^3 +2(0.5) +7 = 8.125
    etc...

    >>>That means, at the negative half of the graph, which is to the left of the y-axis, the graph goes down as x approaches negative infinity.
    Example, if x1 = -0.1, x2 = -0.3, x3 = -0.5, .....
    f(-0.1) = (-0.1)^3 +2(-0.1) +7 = 6.799
    f(-0.3) = (-0.3)^3 +2(-0.3) +7 = 6.373
    f(-0.5) = (-0.5)^3 +2(-0.5) +7 = 5.875
    etc...
    Here, you know that -0.1 is greater than -0.3, and 6.799 is greater also than 6.373. Etc...
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Oct 2005
    Posts
    12
    Thanks ever so much all of you! You are real stars!
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    Some more:

    >>>That means, as x goes from negative infinity to positive infinity, the graph is always going up.

    >>>That means all of the tangent lines to the graph, as we go from left to right, are going up or inclined to the right. That means all tangent lines have positive slopes.

    >>>That means any line segment connecting two points on the graph always has a positive slope.
    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 20th 2011, 12:47 AM
  2. Replies: 3
    Last Post: April 25th 2010, 04:53 PM
  3. Quadratic factors of a cubic polynomial
    Posted in the Pre-Calculus Forum
    Replies: 8
    Last Post: December 10th 2009, 01:01 PM
  4. Replies: 1
    Last Post: June 12th 2008, 10:30 PM
  5. quadratic/cubic funtions expanation
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: November 21st 2007, 09:52 PM

Search Tags


/mathhelpforum @mathhelpforum