# Example of functions satisfying differentiation properties

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.

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Mark44
Mentor
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 = ex - 1.

thanks, that makes sense.

Mark44
Mentor
Items 3 and 4 say that the graph goes through the origin and is concave up.