2

Can I get some examples of f(x) for real x such that:

f(0) = 0 , f(1) = 1

between 0 and 1 exclusive; f'(x) is positive definite

I am looking for different kinds of functions in general (such as x^n)

And any intervals of a function that can be used (such as sin(pi/2*x + 2n*pi) for integer n)

Magic Gonads
  • 23
  • 1
    Let $f$ be any positive continuous function. Then $F(x)=\dfrac{\int_0^x f(t)dt}{\int_0^1 f(t)dt}$ satisfy the given conditions. – mfl Aug 23 '17 at 00:50
  • @mfl Thanks for a general solution! I'm looking for specific forms of functions though.

    Can you explain why this works?

    Also it seems to only need to be positive continuous for positive x, negative x can be ignored, for instance f(x) = log(x+1) works

     – Magic Gonads Aug 23 '17 at 01:14
  • @mfl sin(x) doesn't but sin(x+1) does, and x!-4 does work despite not being a positive function. It seems it doesn't actually need to be a positive function?

    Nevermind sin(x) does work

     – Magic Gonads Aug 23 '17 at 01:19
  • @mfl I see what you mean now, the property that F'(x) is positive definite is only necessarily true for positive f(x): f(x)=x^3 + sin(x+3) does not satisfy this particular condition. – Magic Gonads Aug 23 '17 at 01:26

1 Answers1

0

For positive $a$, let $$f(x)={e^{ax}-1\over e^a-1}$$

Gerry Myerson
  • 179,216