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**Mac "The Man" Smith-XNQBL****Guest**

Hi, im having problems with the following problem. Any help would be great, thanks!

Find the two points on the curve y = x^4 - 2x^2 - x that have a common tangent line.

**Toast****Real Member**- Registered: 2006-10-08
- Posts: 1,321

Find the derivative:

The curve of the derivative has an infinite number of coordinates which share the same 'y' values.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

True, but that wouldn't necessarily mean that they'd have a common tangent. For that, the tangents would need to have the same y-intercept as well as the same gradient.

This is quite a tricky and interesting problem, actually. I'll come back to this in a while.

Why did the vector cross the road?

It wanted to be normal.

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**gnitsuk****Member**- Registered: 2006-02-09
- Posts: 121

therfore

So equation of a tangent to the curve at any point x is:

replacing y from original equation gives: which gives intercept asSo say we have two points which have a common tangent, (a,b) and (c,d) then the x coordinates will have to simultaneously satisfy:

andSo we "just" have to solve these for a and c where a is not equal to b.

Ok, then I "cheat" and use mathematica:

eqns = {4a^3 - 4a - 1 == 4c^3 - 4c - 1, -3a^4 + 2a^2 == -3c^4 - 2c^2}

Solve[eqns, {a, c}]

Whicih gives six solutions including:

and

The others are similarly unpleasant.

This solution gives a is approx. 0.853 and c is approx 0.247 and if you look at the graph of the function at these points:

This looks reasonable.

*Last edited by gnitsuk (2006-10-26 02:22:37)*

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Wow. That was tricky. From the bit where you needed to solve those simultaneous equations, you could do it by writing the second equation in the form (pa²+q)²+r and then writing that in terms of a, then substituting into the first equation and solving for c.

But that would involve a ton of work and Mathematica is a far easier option.

Why did the vector cross the road?

It wanted to be normal.

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