Results 1 to 2 of 2

Math Help - Theorums about Differentiable Functions

  1. #1
    Newbie
    Joined
    Nov 2007
    Posts
    14

    Theorums about Differentiable Functions

    I have 3 questions that I'm stuck on.. if you have any idea about any of them please help me out! Thanks!

    1. Use the fact that ln x and e^x are inverse functions to show that the inequalities e^x >/= 1+x and ln x </= x-1 are equivalent for x>0.

    2. Prove. If f ' (x) </= 1 for all x and f(0)=0, f(x) </= x for all x>/= 0.

    3. Suppose that f ' ' (x) >/= 0 for all x in (a,b). We will show the graph of f lies above the tangent line at (x,f(c)) for any c with a<c<b.
    a) Use the increasing function theorum to prove that f'(c)</=f'(x) for c</= x < b and that f'(x) </= f'(c) for a<x</=c.
    b) Use (a) and the racetrack principle to conclude that f(c) + f'(c)(x-c) </= f(x) for a<x<b.

    Thank you sooooooooooo much
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Paige05 View Post
    I have 3 questions that I'm stuck on.. if you have any idea about any of them please help me out! Thanks!

    1. Use the fact that ln x and e^x are inverse functions to show that the inequalities e^x >/= 1+x and ln x </= x-1 are equivalent for x>0.
    e^x \geq 1+x is equivalent to saying e^x - x - 1 \geq 0 so let f(x) = e^x - x - 1 on (0,\infty) since f'(x) = e^x - 1 and f'(x)>0 it means the function is increasing.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 9
    Last Post: December 17th 2010, 09:13 AM
  2. Differentiable functions.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 17th 2009, 03:47 PM
  3. Set of twice differentiable functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 23rd 2009, 09:36 AM
  4. Sequence of differentiable functions, non-differentiable limit
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 3rd 2009, 06:13 AM
  5. Theorums logic
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: November 30th 2007, 06:58 PM

Search Tags


/mathhelpforum @mathhelpforum