微积分作业代写calclulus代考| Extreme points: The complete story

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微积分作业代写calclulus代考|Remarks

• The difference between Definition $3.1$ and Definition $3.4$ lies in the set of points considered. In Definition $3.1$ only points in the immediate neighbourhood, $S_{r}(\boldsymbol{a})$, of $\boldsymbol{a}$ are considered, while in Definition $3.4$ all points in the domain, $D_{f}$, of the function are involved.
• Definition $3.4$ implies that a critical point, even if a point of local maximum or local minimum, need not be a point of absolute maximum or minimum.

Earlier we said that critical points are examples of extreme points. However, there are other types of extreme points which are not found using the gradient. These are

微积分作业代写calclulus代考|Optimization over compact domains

Recall our definition of a compact set (Definition 1.8): a set $\Omega \subseteq \mathbb{R}^{n}$ is said to be compact if it is closed and bounded.

For any function defined on a region $\Omega \subseteq D_{f}$ that is compact, we have the following very useful result.
Theorem $3.2$
$A$ continuous real-valued function defined on a compact region, $\Omega$, obtains an absolute maximum and an absolute minimum value.
A few comments on this theorem are warranted.
Firstly, it is not necessary that the region being considered is the function’s entire domain of definition, $D_{f}$, but it might be. The problem statement will usually specify this. If no region is given then the reader should assume the whole of $D_{f}$ is implied.

Secondly, by Theorem 1.2, a continuous function defined on a closed and bounded region is necessarily bounded. This means that $|f(\boldsymbol{x})|<K$ for some $K \in \mathbb{R}$ and for all $\boldsymbol{x}$ in that region. This simple result implies that we should expèct $f$ to exhibit an absoolutẻ minimum and an absolutẻ maximum. In fact, this is the only time we are guaranteed that absolute maximum and minimum points exist.

The reader should always bear in mind that a continuous function is not necessarily differentiable everywhere. A consequence of this is that singular points can exist. These should then be inspected separately to any critical points. Naturally, the appealing notion of a closed and finite domain means

that the domain boundary (boundary points) need also to be considered separately.
We illustrate Theorem $3.2$ in action with the following examples.

微积分作业代写calclulus代考|Remarks

• 定义的区别3.1和定义3.4在于所考虑的点集。在定义中3.1只在附近的点，小号r(一种)， 的一种被考虑，而在定义3.4域中的所有点，DF, 的功能都涉及。
• 定义3.4意味着一个临界点，即使是局部最大值或局部最小值，也不必是绝对最大值或最小值的点。