You're confusing the simplified rules (almost always true) with the proper rules (more generally true, but worded more carefully).

\(\displaystyle f'(x)= 0\) means the function is not changing locally (zero slope). This is a requirement for local minima/maxima, but not sufficient. (Look at \(\displaystyle x^3\ @\ x=0\)).

\(\displaystyle f''(x)= 0\) means the slope is not changing locally (flat). If \(\displaystyle f'(a) = 0\text{ and }f''(a) < 0\), then the function is increasing for $x<a$ and decreasing for $x>a$ (in the neighbourhood of a, anyway). This means it MUST be a local maximum. If \(\displaystyle f'(a) = 0\text{ and }f''(a) > 0\), it MUST be a local minimum.

In contrast, \(\displaystyle f'(x)=0\text{ and }f''(x)=0\)

*very often* indicates neither a local minimum nor maximum, but this is not always the case. \(\displaystyle f(x) = x^4\text{ and }f(x) = C\) both prove this.

If the function f(x)=C has both the global minimum and the global maximum equal to C , then this means that the function f(x)=C is both increasing and decreasing âˆ€xâˆˆR?

I think both not decreasing and not increasing might be more accurate.

Would it help if we called $C$ the non-localised global minimum and the non-localised global maximum of \(\displaystyle f(x)=C\)?