Results are out! Find what you need...fast. Get quick advice or join the chat
Hey! Sign in to get help with your study questionsNew here? Join for free to post

differentiable function

Announcements Posted on
Talking about ISA/EMPA specifics is against our guidelines - read more here 05-03-2015
  1. Offline

    ReputationRep:
    1.
    Let
    f(x) = 2x +3 x<=2
    f(x) = x^2 - 2x + 7 x>2

    Show that f'(x) exists at x=2, but f''(2) doesnt exist.



    2.
    Let f,g be functions s.t. the product h(x)=f(x)g(x) is differentiable at a. Does it follow that f and g are differentiable at a?
    is it possible that only 1 of f and g are differentiable at a?


    for 1. do u just diff once and show f'(2)=2 so it exists and f''(2)=0 so it doesnt exist?? if so then y is there a need for 2 functions wen we can fork it out from just the first.
  2. Offline

    ReputationRep:
    1) f'(x) = 2 (x<=2) ->f'(2) = 2
    f'(x) = 2x - 2 (x>2) -> f'(2) = 2.
    So f'(x) at x= 2 exists
    f''(x)= 0 (x<=2)--> f''(2) = 0
    f''(x) = 2 (x>2)-->f''(2) = 2 <> 0.
    So f''(2) doesn't exist.
    2. No, f and g may not be differentiable at a.
    Yes (not so sure).
  3. Offline

    ReputationRep:
    Suppose that f'(a) isnt defined, and g(a) = 0 then we have
    h'(a) = f'(a)g(a) + f(a)g'(a) = f(a)g'(a) which is defined.
  4. Offline

    ReputationRep:
    k i understand 1 but i dnt get numba 2
  5. Offline

    ReputationRep:
    I just answered 2...
  6. Offline

    ReputationRep:
    (Original post by JamesF)
    I just answered 2...
    But not enough, I think. What if both g and f are not differentiable? U just proved that 1 of them is differentiable.
  7. Offline

    ReputationRep:
    :confused: It asks, does the differentiability of h(x) at a imply that both f(x) and g(x) are differentiable at a, and i showed that the answer is no.
    It doesnt take much to extend the arguement to show that neither have to be differentiable at a.
    If g(a) = f(a) = 0, then h'(a) is defined (and equals 0) and neither of f or g need be differentiable at a.
  8. Offline

    ReputationRep:
    (Original post by JamesF)
    :confused: It asks, does the differentiability of h(x) at a imply that both f(x) and g(x) are differentiable at a, and i showed that the answer is no.
    It doesnt take much to extend the arguement to show that neither have to be differentiable at a.
    If g(a) = f(a) = 0, then h'(a) is defined (and equals 0) and neither of f or g need be differentiable at a.
    I might misunderstood the question. I thought it asked if h(x) = f(x).g(x), and was h(x) differentiable if only one of g(x) or f(x) was defined or none of them.
    But ur answer for none of them, I think it's wrong. Let's consider this
    f(x) = (x-1)^1/2 -> f(1) = 0, f'(1) is not defined.
    g(x) = (x-1)^1/3 -> g(1) = 0, g'(1) is not defined.
    h(x) = (x-1)^5/6 -> h'(x) = 5/6(x-1)^(-1/6) which is not defined at x = 1.
    But quite similarly, ur answer is right if g(x) = (x-1)^2/3. I mean ur answer for neither of them is differentiable can't be generalization.
Updated: January 24, 2005
2015 general election
New on TSR

Vote in the TSR Political Party Contest!

Choose which TSR Party you want in Number 10

Article updates
Quick reply
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.