T or F: If a graph of a function has a tangent line at some point (a, b) on the curve and another tangent line

Question

...at point (c, d), also on the curve, then the two tangent lines can never intersect.

Answer 1

False.

Imagine your function describes a semicircle. Every tangent line on that semicircle will intersect with the others if they are extended far enough. Draw one out if you can't imagine it in your head.

Answer 2

Definitely false. In fact, the function has little role to play here. Since tangent lines are just lines in the plane, any two _arbitrary_ lines will very likely intersect.

Now, if the question stated that the function has equal slope at points a and c, then the tangent lines would be parallel, in which case they would not intersect. But the question didn't say this.

Answer 3

False. The tangent lines will usually intersect. However, if the lines drawn tangent to the curve are parrallel, then they (obviously) will not intersect. Tangent lines drawn to a parobola will intersect, but if you are dealing with a hyperbola, circle or ellipse then it is possible to have lines of tangency that will not intersect.

Answer 4

no. it is not true. take the case of parabola, any two tangents intersect

Answer 5

I failed math. It's a.m. You just made me think and I now have a huuuuuuuuuge headache. Hope you're happy!