Sample+Question+Differentiability

1. there is a distinct possibility that f"(a) could not exist. For example, in certain piecewise functions such as f(x) = x^2 as x ≥ 0, -x^2 as x≤0, the first derivative would be f'(x) = 2x as x ≥ 0, -2x as x≤0, however the second derivative of that equation is as follows f"(x) = 2 as x ≥ 0, -2 as x≤0, which is undefined. - Zach Fusfeld

A) True because if f(x) is differentiable at x = a then that means there is no corner point or cusp at x = a, however, it can mean that x = 0 and the second derivative does not exist B) True: As you approach a the from the right and left will be equal meaning that it is differentiable

C) Implies that the derivative of the f(x) exists as x approaches a which would be true if the f(x) is differentiable because by saying it is differentiable you are implying that x = a does exist (True)

D) Is also True for the same reasons as C

E) FALSE: it is interminably if the second derivative is true because the first derivative can exist without the second derivative existing, for the same reasons Zach stated. For Example: If f '(x) = 0 then f "(x) would be undefined

-Marnie Wachs