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    I came the following difficult question.
    The curve C with equation y=x3+3x2-3x is reflected in the line y=x onto the curve C'.
    (a) Find the equation of the curve C'.
    (b) Find the equation of the line L, which is tangent to both curve C and curve C'.
    [Your equation for L should be in its exact form. No credit will be given for any solutions found using differentiation.]
    This is actually a difficult question. Any ideas?
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    (Original post by Ano123)
    I came the following difficult question.
    The curve C with equation y=x3+3x2-3x is reflected in the line y=x onto the curve C'.
    (a) Find the equation of the curve C'.
    (b) Find the equation of the line L, which is tangent to both curve C and curve C'.
    [Your equation for L should be in its exact form. No credit will be given for any solutions found using differentiation.]
    This is actually a difficult question. Any ideas?
    Wth why would they ban differentiation what's the point haha. What level is that question from? Also Is there a diagram for it?
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    (Original post by Nayzar)
    Wth why would they ban differentiation what's the point haha. What level is that question from? Also Is there a diagram for it?
    It's FP1, there is a diagram. But I haven't taken a picture of the question. I'll upload one.
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    (Original post by Ano123)
    This is actually a difficult question. Any ideas?
    I think that the biggest hint that I can give you is that the question is much easier than you seem to think it is. Another hint in the spoiler:
    Spoiler:
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    Think about the relationship between the line L and the line y = x.
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    (Original post by Gregorius)
    I think that the biggest hint that I can give you is that the question is much easier than you seem to think it is. Another hint in the spoiler:
    Spoiler:
    Show
    Think about the relationship between the line L and the line y = x.
    It has gradient of -1 right? I know that and have seen questions like this before. But why does it have to be -1? What's an intuitive way of understanding this?
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    (Original post by Ano123)
    What's an intuitive way of understanding this?
    Look for symmetries.
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    Does anyone have an answer?
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    (Original post by Ano123)
    Does anyone have an answer?
    I could be missing a trick but the answer isn't at all obvious to me.

    If we zoom out a bit, then we can see there are two lines meeting the question's criterion.


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    And the algebra was a bit tortourous (and I got it wrong!); looking for a repeated root in the equation -x+c =x3+3x2-3x


    And if we change the equations slightly, by adding 3 say, then there are even more solutions and the additional ones are not perpendicular to the line of symmetry:

    Attachment 545941545943
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    (Original post by Ano123)
    Does anyone have an answer?
    Sorry for delay; got called away. It's clear from symmetry that the slope of the line is -1. So the equation of the line can be expressed as  y = \alpha - x. We need to find \alpha. This line is tangent to the original curve, so we need to find \alpha such that x^3 + 3x^2 -3x = \alpha - x has a double root at some value of x = a. (It is tangent there and intersects the curve for some smaller value of y not plotted on the original graph). So, solve the equation x^3 + 3x^2 -2x -\alpha = (x-a)^2(x-b) for the value of a. You should find that the same equation that calculus will give drops out!
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    (Original post by ghostwalker)
    I could be missing a trick but the answer isn't at all obvious to me.

    If we zoom out a bit, then we can see there are two lines meeting the question's criterion.


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    PRSOM. I assumed that the picture given was part of the question and therefore the line shown was what was required. Then the algebra reduces to finding a double root (with the line intercepting the original curve, as you have shown, for a more negative value of y).
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    Thank you for your help. I did get the answer but I used the discriminant and set it equal to 0 for the intersection of y=c-x and found c by solving the quadratic formed in c. As ghostwalker said there are actually 2 lines, but L is indicate don the diagram. The repeated root method is a nice one as well but I didn't think to do this.
    There's nothing you lot can't do it seems.
 
 
 
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