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[itex]x^4+c*x^3+12x^2-5x+2[/itex]

There can always be a straight line intersecting four points while the bottom two 'humps' have a common tangent line. I'm looking to find the value of c where that left hump disappears, which can be found when there are no two points that share a common tangent line.

Starting now, finding the general tangent equation:

[itex]y=(4a^3+3*c*a^2+12a-5)*x+b[/itex] Note, a is the x value that the tangent is built from

Since both the tangent and the equation share one known point (where x=a), we can find b

[itex]x^4+c*x^3+12x^2-5x+2=(4a^3+3*c*a^2+12a-5)*x+b[/itex]

[itex]a^4+c*a^3+12a^2-5a+2=4a^4+3*c*a^3+12a^2-5a+b[/itex]

[itex]b=-3a^4-2*c*a^3-12a^2+2[/itex]

This gives us the general equation of:

[itex]y=(4a^3+3*c*a^2+24a-5)x-3a^4-2*c*a^3-12a^2+2[/itex]

On the two points that share a common tangent line, the height of both the tangent line and the original function will be the same. This allows us to solve for x:

[itex]x^4+c*x^3+12x^2-5x+2=(4a^3+3*c*a^2+24a-5)x-3a^4-2*c*a^3-12a^2+2[/itex]

[itex]x^4+c*x^3+12x^2-5x-(4a^3+3*c*a^2+24a-5)x+3a^4+2*c*a^3+12a^2=0[/itex]

The four roots of this equations are:

x and a are both on the bottom 'hump'

x and a are both on the top 'hump'

x is on the bottom 'hump' and a is on the top

x is on the top 'hump' and a is on the bottom

For the first two options, x=a. So the last equation can be divided by x=a twice.

[itex]\frac{x^4+c*x^3+12x^2-5x-(4a^3+3*c*a^2+24a-5)x+3a^4+2*c*a^3+12a^2}{(x-a)^2}=0[/itex]

[itex]x^2+(2a+c)x+3a^2+2*c*a+12=0[/itex]

Solving for x yields:

[itex]x=\frac{-2a-c\pm SQRT(c^2-4*c*a-8a^2-48)}{2}[/itex]

This formula finds x (the second point that shares a common tangent line) in terms of the first point that shares the common tangent line. The square root allows us to find where a common tangent line is shared, where the stuff inside the square root is less than 0, no real value for x exists, meaning there is no second point that shares a common tangent line.

[itex]c^2-4*c*a-8a^2-48>0[/itex]

[itex]c>\frac{4a\pm SQRT(16a^2+32a^2+192)}{2}[/itex]

[itex]c>2a\pm 2*SQRT(3a^2+12)[/itex]

From here, I'm stuck