A tangent line to both functions

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Homework Help Overview

The problem involves finding a line that is tangent to two functions, f(x) = x² and g(x) = x² - 2x. Participants are exploring the conditions under which a single line can be tangent to both curves.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to derive relationships between the derivatives of the two functions to find points of tangency. Some participants question the validity of the relationships established and explore the implications of those equations.

Discussion Status

The discussion is ongoing, with participants providing insights and corrections to each other's reasoning. There is a focus on clarifying the relationships between the variables involved in the tangent line equations.

Contextual Notes

Participants are working under the constraints of the problem statement and are addressing potential misunderstandings in the mathematical relationships derived from the functions.

Atran
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Homework Statement



Determine a line that is tangent to both f(x)=x2 and g(x)=x2-2x

Homework Equations



The Attempt at a Solution


f(x)=x2 => f'(x)=2x
g(x)=x2-2x => g'(x)=2x-2

f'(a) = f'(b)
2a = 2b-2

I don't know how to continue.
Thanks for help.
 
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If h(x) is that line, what do you know about h(a), h(b)?
 
Thank you.

h(b) = (2b-2)b+c => c = -b2

h(a) = (2b-2)a+c = (2b-2)a-b2 => (a-b)2+2a=0
2a = 2b-1 => a = b-1
(-1)2+2a=0 => a=-0.5 and b=0.5
 
2a = 2b-1 => a = b-1
That is not true, but I think you mean 2(b-1).
The result is right.
 
Yes, I meant 2b-2.
 

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