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Math Help - Find the cubic function y = x3 + ax2 + bx + c

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    Find the cubic function y = x3 + ax2 + bx + c

    Find the cubic function y = x^3 + ax^2 + b^x + c whose graph has a horizontal tangent at (-2, 10) and passes through (1,1).
    What I tried so far is plugging in -2 and 1 in the original function to get:
    -8 +4a -2b +c = 10
    1 + a + b + c = 1

    And I have y' = 3x^2 + 2ax + b, which when I plug in -2 I get
    0 = 12 - 4a + b (so b = -12 + 4a), but I have no clue where to go from there..even though I have a feeling it's something easy.
    So if someone could give me a hint as to where to go next, that would be great! thanks for your time
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    Re: Find the cubic function y = x3 + ax2 + bx + c

    Quote Originally Posted by aimforthehead View Post
    Find the cubic function y = x^3 + ax^2 + b^x + c whose graph has a horizontal tangent at (-2, 10) and passes through (1,1).
    What I tried so far is plugging in -2 and 1 in the original function to get:
    -8 +4a -2b +c = 10
    1 + a + b + c = 1

    And I have y' = 3x^2 + 2ax + b, which when I plug in -2 I get
    0 = 12 - 4a + b (so b = -12 + 4a),
    Well, you have three equations in three unknowns.
    \begin{align*}4a-2b+c&=18\\a+b+c &=0\\4a-b &=12   \end{align*}
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    MHF Contributor MarkFL's Avatar
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    Re: Find the cubic function y = x3 + ax2 + bx + c

    You're off to a good start. I would write the 3 resulting equations as:

    (1) 4a-2b+c=18

    (2) a+b+c=0

    (3) 4a-b=12

    Now, if you subtract (2) from (1), you get:

    3a-3b=18 or a-b=6

    Now, if you subtract this from (3) you get:

    3a=6 or a=2 and this means 2-b=6\,\therefore\,b=-4

    and then substituting into (2) we find 2-4+c=0\,\therefore\,c=2
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    Re: Find the cubic function y = x3 + ax2 + bx + c

    I can't believe I forgot how to find three unknowns from three equations >.>
    Well thanks for clearing it up, after that and a quick tutorial from Khan Academy I was quickly able to find the solution!
    It turned out to be:
    y = x^3 + 2x^2 - 4x + 2
    Also thanks to the person above me for working it out, now I'm certain it is correct
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