CheemnCatelvew
2021-06-11
Answered

Find all real solutions of the equation, rounded to two decimals.

${x}^{4}-8{x}^{2}+2=0$

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coffentw

Answered 2021-06-12
Author has **103** answers

Step 1

To find: all real solutions of the equation, rounded to two decimals.

Let

Step 2

if

therefore,

and the solutions are

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Suppose $\{{f}_{n}\}$ is a sequence of (Lebesgue) measurable functions on ${\mathbb{R}}^{d}$ with each ${f}_{n}\ge 0$. If ${f}_{n}(x)\to f(x)$ for a.e. $x$, then by Fatou's lemma, we have

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Its proof begins with a non-negative function $g$ that is bounded and supported on a set $E\subseteq {\mathbb{R}}^{d}$ of finite measure with $g\le f$. If we set ${g}_{n}=min\{g,{f}_{n}\}$, then each ${g}_{n}$ is a measurable function and supported on $E$. Furthermore, ${g}_{n}(x)\to g(x)$ for a.e. $x$. By the bounded convergence theorem, ...

The above material is quoted from the book by Stein and Shakarchi with some minor changes, and I wonder why ${g}_{n}\to g$ almost everywhere. I wish I could offer you some useful ideas, but unfortunately I know nothing, which is why I'm here. I would much appreciate it if you could do me a favor. Thank you.

$\int f\le \underset{n\to \mathrm{\infty}}{lim\u2006inf}\int {f}_{n}.$

Its proof begins with a non-negative function $g$ that is bounded and supported on a set $E\subseteq {\mathbb{R}}^{d}$ of finite measure with $g\le f$. If we set ${g}_{n}=min\{g,{f}_{n}\}$, then each ${g}_{n}$ is a measurable function and supported on $E$. Furthermore, ${g}_{n}(x)\to g(x)$ for a.e. $x$. By the bounded convergence theorem, ...

The above material is quoted from the book by Stein and Shakarchi with some minor changes, and I wonder why ${g}_{n}\to g$ almost everywhere. I wish I could offer you some useful ideas, but unfortunately I know nothing, which is why I'm here. I would much appreciate it if you could do me a favor. Thank you.