I can see that the definition of local maximum and unconstrained local maximum is written differently, but to me they look like they are defining the same thing. Furthermore, based on Fig 4.1, it looks like both $x^*$ and $y^*$ meet the definition of local maximum and unconstrained local maximum?
How do I distinguish between the 2 definitions?
I'll assume $\mathcal{D}$ refers to the domain of the function.
I'm not familiar with the term "unconstrained local maximum," but the definitions given here are different. Look at the left-most point on the curve; say that it's located at $x = x_0$. Then, taking a small ball $B(x_0, \varepsilon)$, $f(x_0) > f(y)$ for all $y \in B(x_0, \varepsilon) \cap \mathcal{D}$ so $x_0$ is a local maximum by this definition.
On the other hand, no ball $B(x_0, r)$ is contained in $\mathcal{D}$ -- $\mathcal{D}$ contains no points to the left of $x_0$ -- so $x_0$ cannot be an unconstrained local maximum by this definition.
So, in other words, according to these definitions an "unconstrained local maximum" is a "local maximum" which occurs in the middle of the domain, as opposed to at an endpoint.
