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| Math Central | Quandaries & Queries |
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I have this problem I have been working on for days and cannot figure it out. 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? |
Chris,
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.
Chris
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