Question

Question: Is it true that if f(x) is a function and L is the line with equation y= -1
then L cannot be tangent to the graph of f(x) at more than one point?

Answer 1

False,
here is a counter example
f(x) = (x+1)^2(x-1)^2 -1
this has 2 local minimums at y=-1 , thus tangent to f(x) at 2 points

http://www.wolframalpha.com/input/?i=y+%3D+%28x%2B1%29^2*%28x-1%29^2+-1%2C+y%3D-1

Answer 2

Another good counter example is sinx or cosx.