My textbook has these passages written and I'm a bit confused as to whether local maximums and minimums can be on endpoints. why or why not?
Apparently x=4 is not a local maximum. But why?
Other passages:
2 Answers
$\begingroup$can local maximums and minimums be on endpoints?
Not according to the definition in your second scanned image.
Apparently $x=4$ is not a local maximum. But why?
Short answer: because the "near $c$" language in the definition means "near $c$" on both sides of $c$.
Longer answer: According to the definition in the red box of your second scanned image,
the number $f(c)$ is a local maximum value of $f$ if $f(c) \geq f(x)$ when $x$ is near $c$
where "near $c$" means
on some open interval containing $c$.
However, for the function $f$ whose graph is plotted in Figure 7, every open interval containing $4$ will contain some values $x > 4$, for which $f(x)$ is not defined. (The domain of $f$ was given to be the closed interval $[-1, 4]$.) So, it's not possible to have $f(4) \geq f(x)$ for every $x$ in some open interval containing $4$.
The discussion above addresses the definition of local maximum, but the definition for local minimum is analogous and has the same issue.
$\endgroup$ $\begingroup$When you look for optima of a function on a bounded interval (as opposed to the entire space), the optimum point can come from
- Local extrema inside the boundary
- Boundary itself.
A good example is $f(x) = x^2$ on $I=[-1,2]$ which has a unique minimum on $\mathbb{R}$, which happens to fall inside $I$, but for the maximum, the situation is more tricky. $f$ has no local maximum over $\mathbb{R}$ but the boundary is limited to $2$ points $(-1,1)$ and $(2,4)$, both of which are candidates for the optimum and indeed it occurs at $(2,4)$.
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