The functions f(x) is increasing on a domain D if a) f'(x) < 0 for all x in D b) f'(x) = 0 for all x in D c) f'(x) 0 for all x in D d) f'(x) doesn't exist for all x in D

A function $f(x)$ is said to be increasing on a domain $D$ if its derivative satisfies a specific condition. By definition, a function is increasing on an interval if its derivative is positive for all $x$ in that interval.

• If $f^{\prime }(x)>0$ for all $x\in D$, then $f(x)$ is strictly increasing on $D$.
• If $f^{\prime }(x)\ge 0$ for all $x\in D$ (and not identically zero on any subinterval), then $f(x)$ is non-decreasing (weakly increasing).

The other options are incorrect because:

• $f^{\prime }(x)<0$ implies the function is decreasing.
• $f^{\prime }(x)=0$ implies the function is constant (not increasing).
• $f^{\prime }(x)$ not existing does not guarantee an increasing function.

Thus, the correct answer is (c) $f^{\prime }(x)>0$ for all $x\in D$.