Math CentralQuandaries & Queries


I have this problem I have been working on for days and cannot figure it out.
it states: Find the two points on the curve y=x^4 - 2x^2 - x that have a common tangent line.

I know you use the first derivative to find the tangent line so if it is a common tangent line should you find two of the exact same tangent line equations at two coordinate points?


Your method sounds way too hard. Why not simply guess what the answer must be, taking as a hint the fact that x4 - 2x2 - x is "almost" a perfect square in the variable u = x2 -- you just have to change the -x into a +1. If you want the line y = mx + b to be tangent twice to the given curve, the equation that results by plugging y= mx + b in for y will have to factor into a product of 2 perfect squares in x. So try letting
y = -x - 1.
You can then solve the resulting fourth degree equation for x to find the two values of x where the line
y = -x-1 is tangent.


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