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$f(x)=1/x; x>0$ 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 $f(2)<f(x)$ 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?

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$f(1), f(2)$ 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.

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  • $\begingroup$ Thank you very much $\endgroup$
    – user163862
    Nov 16, 2021 at 0:17

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