Question

Okay last question I swear. Another law question with derivatives. Problem in comments

Answer 1

if the limit of a function, the limit of the difference quotient, and the function itself exist and are the same, the function is continuous, differentiable, and defined.

Answer 2

the only way to know if III is true is to knwo about the second derivative. But you don't have information on that.

Answer 3

So how do I check the other two? That's mostly what I was stuck on

Answer 4

Okay, well, what's f(2) equal?

Answer 5

Isn't f(2) that equation?

Answer 6

omg I am so sorry. Okay. Ignore what i said before. What you need to know is that if a function is differentiable, it must be continuous. However, you dont' know anything abou thte derivative of a derivative.

Answer 7

Okay the whole question just seems dumb to me lol. Thanks for your help

Answer 8

Honestly, this thing has me stumped. If you have that limit set up equal to some value, then it seems like the derivative exists, which means it is differentiable at x = 2.

If it's differentiable at some value, then it has to be continuous on f(x) at that the same value. So II implies I simply based on how derivatives are constructed (with the limit definition).

If f(x) is differentiable at x = c, then f ' (x) is continuous at x = c. If you have some jump in value on f ' (x) at x = c (or near it), then you have a sudden change in slopes which leads to sharp undifferentiable points. So if II is true, then III must be true as well.

That's why I'm thinking all 3 (I, II, III) are true. But that choice isn't listed. It's just odd.

Answer 9

jim I thought it would just be I and II

Answer 10

if II is true, then III has to be true

Answer 11

you can't have f(x) differentiable at x = c and have f ' (x) discontinuous at x = c.

Answer 12

and if III is true, then II has to be true

Answer 13

I thought you could... you can't have f(x) differentiable at x=c and have f(x) discontinuous at x=c, but you can have what you mentionioed?

Answer 14

yeah f(x) also needs to be continuous at x = c as well

Answer 15

a good example is f(x) = |x|

f(x) is continuous, but f ' (x) has a discontinuity at x = 0, so f(x) is not differentiable at x = 0.