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Homework Help: 2 Calculas Questions

  1. Aug 12, 2010 #1
    Question 1:

    1. The problem statement, all variables and given/known data

    Find "b" so that the difference between the x co-ordinates of the two inflection points of y=x4 + 4x3+bx2+5x+7 is 3

    3. The attempt at a solution

    I.P. at y' = 0
    two x co-ordinates are x and x+3
    y' = 4x3 + 12x2 + 2bx +5
    0 = 4x3 + 12x2 + 2bx +5
    0 =4(x+3)3 + 12(x+3)2 + 2b(x+3) +5

    4x3 + 12x2 + 2bx +5 = 4(x+3)3 + 12(x+3)2 + 2b(x+3) +5

    expand, add, subtract and solve for x, you get:

    (-216-6b)/72 = x

    I.P is y''=0

    y'' = 12x2 + 24x +2b
    sub in x

    0= 12[(-216-6b)/72]2+24 (-216-6b)/3]+2b

    then you expand, add, subtract and end up with the quadratic equation

    0= 36 (b2+48b +432)

    use the quadratic formula to get b=-12 or or b=-36

    to check if b is right, I plugged it back in to y''

    0= 12x2 + 24x -2b
    0=12x2 + 24x -2(-12)
    0 = 12x2 + 24x -24
    x=-3.8284 or x=1.8284
    but the difference here is 5.6568

    0=12x2 + 24x -2(-36)
    x= -1, x = 3
    the difference here is 4.

    Question 2:

    1. The problem statement, all variables and given/known data

    if f(x) = x3 + 3x2 + k has three distinct real roots, what are the bounds on "k"? (i.e., ? <x<?). Hint: Look for extema using f' and f''.

    3. The attempt at a solution

    f(x) = x3 + 3x2 + k
    f'(x)= 3x2 +6x
    0= 3x(x+2)
    x = 0, x=-2

    f"(x) = 6x+6
    f'' (0) = 6

    at x=0, the function is concave up

    f''(-2) = -6

    at x = -2, it is concave down

    because of the type of graph (its a cubic function), the y-int (which is k) must be equal to or less than 0, but has to be greater than equal to -4...

    as I was typing this, I realized that this doesnt make sense, so I dont understand this question.

    I will appreciate any help I can get.
     
  2. jcsd
  3. Aug 12, 2010 #2

    ehild

    User Avatar
    Homework Helper


    The inflection points means y''=0.

    The result is almost good, but I do not understand the explanation. Sketch the graph. It has three different real roots so it has three x points where y change from negative to positive or vice versa. It has two extrema, one at x=-2, one at x =0. What are these extrema, where is minimum and where is maximum? Where should be the function positive and where is it negative?

    ehild
     
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