Example of functions satisfying differentiation properties

[kathrynag]

Suppose the function
f has the following four properties:
1. f is continuous for x >=0;

2.
f'(x) exists for x > 0;

3.
f(0) = 0;

4.
f'is monotonically increasing.

I'm just looking for functions that have these 4 properties to better understand what f represents.
So far, I came up with x^2 and x^3, but was looking for more examples. I'm just looking for examples so I can graph these and see a graphical representation. I was getting stuck on good functions to use.

 

[Mark44]

Suppose the function
f has the following four properties:
1. f is continuous for x >=0;

2.
f'(x) exists for x > 0;

3.
f(0) = 0;

4.
f'is monotonically increasing.

I'm just looking for functions that have these 4 properties to better understand what f represents.
So far, I came up with x^2 and x^3, but was looking for more examples. I'm just looking for examples so I can graph these and see a graphical representation. I was getting stuck on good functions to use.

How about exponential functions, translated so that they go through the origin? E.g., y = e^x - 1.

 

[kathrynag]

thanks, that makes sense.

 

[Mark44]

Items 3 and 4 say that the graph goes through the origin and is concave up.