# Max or min values on closed interval of $f(x)=1/x; x>0$

$$f(x)=1/x; x>0$$ The question is does this function have a max or min on the interval $$[1,2]$$.

Since the $$x$$ values of $$1$$ and $$2$$ are in the domain and I can evaluate that $$f(1)> f(x)$$ for all $$x$$ in the function's closed interval AND I can also evaluate that $$f(2) for all of the function's values in the closed interval, then can I say that $$f(1)$$ is a maximum value and that $$f(2)$$ is a minimum value?

Because they are endpoints are they NOT considered global max and min values?

$$f(1), f(2)$$ are the global max and global min in the interval $$[1,2]$$. The phrase (for example) global min is intended to provide a contrast to the phrase local min, which would signify that there is some neighborhood within $$[1,2]$$, such that for a specific value $$x_0$$ in this neighborhood, $$f(x_0)$$ is the min value for those values of $$x$$ in this neighborhood.
For example, with a similar but different function on $$[1,2]$$, there might be the interval $$(1.4, 1.6)$$ such that $$f(x)$$ achieves a minimum value in this neighborhood at $$f(1.5)$$.
So, in my altered hypothetical, if the overall minimum value for $$f(x)$$ on $$[1,2]$$ is $$f(2)$$, then $$f(1.5)$$ would be a local minimum and $$f(2)$$ would be the global minimum.