I am looking for a non-linear function f(x) such that f'(x1) = f'(x2) but x1 != x2.
I guess sin(x) will do... :)
Although I wouldn't mind hearing other suggestions, I'm sure there are lots of functions that satisfy this requirement.
Any function that is not injective has the quality that f(x_1) = f(x_2) does not imply that x_1 = x_2.
edited to add: oops I see he little ' now,, in this case you just need it so the derivative of the function is not injective.
So quickly f'(x) = 3x^2 is not injective and f(x) = x^3 is non-linear.
Here's a couple :
1) any periodic function,
2) any polynomial higher than a quadratic
Hmm, sorry I think I forgot to mention one thing. I tried to simplify the task, which is to find a non-linear function, so that its tangent line at (x1, f(x1)) is the same tangent line at (x2, f(x2)). So in that perspective, x^3 doesn't work.
Separate names with a comma.