<s>[INST] Extreme points of a function at domain ends Consider the following function: $f(x) = x\sqrt{9-x^2}$
$\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad $
$f'(x) = \frac{-2x^2+9}{\sqrt{9-x^2}}$ and $D(f) = [-3,3]$ therefore the critical points of the function are $x_{c_i} = \left\{ -3, -\frac{3\sqrt2}{2}, \frac{3\sqrt2}{2}, 3 \right\}$

Apparently the points $\{-\frac{3\sqrt2}{2}, \frac{3\sqrt2}{2}\}$ are global minumum and global maximum respectively.
But what about the domain ends $\{-3, 3\}$? Are they considered to be saddle points, local minimums, or local maximums and why?
 [/INST] If $D(f)=[-3,3]$, then $f( \pm 3)=0.$
Let $x \in (-3,0)$, then $f(x)<0=f(-3)$, henc $f$ has a local maximum at $x=-3.$
Let $x \in (0,3)$, then $f(x)>0=f(3)$, henc $f$ has a local minimum at $x=3.$
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