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For what values of c is there a straight line that intersects the curve in four distinct places?
[itex]x^4+c*x^3+12x^2-5x+2[/itex]
I'm looking for a full answer (doesn't have to use the same method)
[itex]x^4+c*x^3+12x^2-5x+2[/itex]
This is what the equation looks like where c=7 (red) and c=6 (blue), to help get an idea.
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
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