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By admin 数学代写 2022年2月22日

Now solve the result for $x$. The process by which this was accomplished in adding in the term $b^{2} / 4 a^{2}$ is referred to as completing the square. You should obtain the quadratic formula,
$$
x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}
$$
The expression $b^{2}-4 a c$ is called the discriminant. When it is positive there are two different real roots. When it is zero, there is exactly one real root and when it equals a negative number there are no real roots.
Find $u$ such that $-\frac{b}{2}+u$ and $-\frac{b}{2}-u$ are roots of $x^{2}+b x+c=0$. Obtain the quadratic formula from this.
Suppose $f(x)=3 x^{2}+7 x-17$. Find the value of $x$ at which $f(x)$ is smallest by completing the square. Also determine $f(\mathbb{R})$ and sketch the graph of $f$. Hint:
$$
\begin{aligned}
f(x) &=3\left(x^{2}+\frac{7}{3} x-\frac{17}{3}\right)=3\left(x^{2}+\frac{7}{3} x+\frac{49}{36}-\frac{49}{36}-\frac{17}{3}\right) \
&=3\left(\left(x+\frac{7}{6}\right)^{2}-\frac{49}{36}-\frac{17}{3}\right)
\end{aligned}
$$
Suppose $f(x)=-5 x^{2}+8 x-7$. Find $f(\mathbb{R})$. In particular, find the largest value of $f(x)$ and the value of $x$ at which it occurs. Can you conjecture and prove a result about $y=a x^{2}+b x+c$ in terms of the sign of $a$ based on these last two problems?
Show that if it is assumed $\mathbb{R}$ is complete, then the Archimedean property can be proved. Hint: Suppose completeness and let $a>0$. If there exists $x \in \mathbb{R}$ such that $n a \leq x$ for all $n \in \mathbb{N}$, then $x / a$ is an upper bound for N. Let $l$ be the least upper bound and argue there exists $n \in \mathbb{N} \cap[l-1 / 4, l]$. Now what sbout $n+1$ ?
Suppose you numbers $a_{k}$ for each $k$ a positive integer and that $a_{1} \leq a_{2} \leq \ldots .$ Let $A$ be the set of these numbers just described. Also suppose there exists an upper bound $L$ such that each $a_{k} \leq L$. Then there exists $N$ such that if $n \geq N$, then $\left(\sup A-\varepsilon<a_{n} \leq \sup A\right] .$
If $A \subseteq B$ for $A \neq \emptyset$ and $A, B$ are sets of real numbers, show that inf $(A) \geq \inf (B)$ and $\sup (A) \leq \sup (B)$.
2.13 The Complex Numbers
Just as a real number should be considered as a point on the line, a complex number is considered a point in the plane which can be identified in the usual way using the Cartesian coordinates of the point. Thus $(a, b)$ identifies a point whose $x$ coordinate is $a$ and whose $y$ coordinate is $b$. In dealing with complex numbers, such a point is written ass $a+i b$. For example, in the following picture, I have graphed the point $3+2 i$. You see it corresponds to the point in the plane whose coordinates are $(3,2)$.

现在求解 $x$ 的结果。通过添加术语 $b^{2} / 4 a^{2}$ 来完成此过程的过程称为完成正方形。你应该得到二次公式， $$ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} $$ 表达式 $b^{2}-4 a c$ 称为判别式。当它为正时，有两个不同的实根。当它为零时，只有一个实根，当它等于负数时，没有实根。 17. 求 $u$ 使得 $-\frac{b}{2}+u$ 和 $-\frac{b}{2}-u$ 是 $x^{2}+b x+c= 的根0 美元。从中得到二次公式。 18. 假设 $f(x)=3 x^{2}+7 x-17$。通过完成平方找到 $f(x)$ 最小的 $x$ 的值。还要确定 $f(\mathbb{R})$ 并画出 $f$ 的图形。暗示： $$ \开始{对齐} f(x) &=3\left(x^{2}+\frac{7}{3} x-\frac{17}{3}\right)=3\left(x^{2}+\frac {7}{3} x+\frac{49}{36}-\frac{49}{36}-\frac{17}{3}\right) \\ &=3\left(\left(x+\frac{7}{6}\right)^{2}-\frac{49}{36}-\frac{17}{3}\right) \end{对齐} $$ 19. 假设 $f(x)=-5 x^{2}+8 x-7$。求 $f(\mathbb{R})$。特别是，找到 $f(x)$ 的最大值和它出现的 $x$ 的值。你能根据最后两个问题用$a$ 的符号来推测和证明$y=a x^{2}+b x+c$ 的结果吗？ 20. 证明如果假设$\mathbb{R}$ 是完备的，那么可以证明阿基米德性质。提示：假设完整性并让$a>0$。如果存在 $x \in \mathbb{R}$ 使得 $na \leq x$ 对于所有 $n \in \mathbb{N}$，则 $x / a$ 是 N 的上界。设 $l $ 是最小的上界并且认为存在 $n \in \mathbb{N} \cap[l-1 / 4, l]$。现在怎么样了 $n+1$ ？ 21. 假设你为每个 $k$ 一个正整数编号 $a_{k}$ 并且 $a_{1} \leq a_{2} \leq \ldots .$ 让 $A$ 是刚刚描述的这些数字的集合.还假设存在一个上限$L$，使得每个$a_{k} \leq L$。那么存在 $N$ 使得如果 $n \geq N$，那么 $\left(\sup A-\varepsilon