Question

Given that the limit as h approaches 0 of (f(6 + h) - f(6))/h = -2, which of these statements must be true?

1. f'(6) exists
2. f(x) is continuous at x=6
3. f(6) < 0

Answer 1

that is the very definition of the derivative of f(x) at x = 6
so
it exists, it is -2
the function is contiuous because it has a unique slope at 6
All I know is that the SLOPE is negative 2 at x = 6. I do NOT KNOW if the function is +, - or 0

Answer 2

Isn't #2 also supposed to be true? I thought f(x) had to be continuous at the point at which we find the derivative.

Answer 3

I agree :)

Answer 4

I suppose you could have the function jump up or down at x = 6 and have the same slope of -6 on both sides of the jump, but that is a bit of a stretch.