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$\frac{20b}{{\left(4b^3\right)}^3}$

1. A body falls from rest against resistance proportional to the square root of the speed at any instant. If the body attains speed V1 and V2 feet per second, after 1 and 2 seconds in motion, respectively, find an expression for the limiting velocity.

If $f(x)=ax-<!--\; -\; -->\frac{x^3}{1+x^2}$ show that $\mathrm{\forall <!--\; \forall \; -->}x[f^{\prime }(x)\ge <!--\; \ge \; -->0$ iff $a\ge <!--\; \ge \; -->\frac{9}{8}]$
Part of answer found online:
$f^{\prime }(x)=a-<!--\; -\; -->\frac{3x^2(1+x^2)-<!--\; -\; -->2x(x^3)}{(1+3x^2{)}^2}=a+\frac{x^4-<!--\; -\; -->3x^2}{(1+x^2{)}^2}$
My problem:
I don't understand why:
$a-<!--\; -\; -->\frac{3x^2(1+x^2)-<!--\; -\; -->2x(x^3)}{(1+3x^2{)}^2}=a+\frac{x^4-<!--\; -\; -->3x^2}{(1+x^2{)}^2}$
When i try to rearrange the left i get:
$a-<!--\; -\; -->\frac{3x^2(1+x^2)-<!--\; -\; -->2x(x^3)}{(1+3x^2{)}^2}=a-<!--\; -\; -->\frac{3x^4+3x^2-<!--\; -\; -->2x^4}{(1+3x^2{)}^2}=a-<!--\; -\; -->\frac{x^4+3x^2}{(1+3x^2{)}^2}$
And from there:
$a-<!--\; -\; -->\frac{x^4+3x^2}{(1+3x^2{)}^2}=a+\frac{-<!--\; -\; -->1}{1}\cdot <!--\; \cdot \; -->\frac{x^4+3x^2}{(1+3x^2{)}^2}=a+\frac{-<!--\; -\; -->x^4-<!--\; -\; -->3x^2}{(1+3x^2{)}^2}$
Am i making a simple mistake in the interpretation of the minus in front of a fraction?

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Is a horizontal line an increasing or decreasing function?
This is the definition of an increasing and decreasing function.
"A function f(x) increases on an interval I if $f(b)\ge f(a)\mathrm{\forall <!--\; \forall \; -->}b>a$, where $a,b\in <!--\; \in \; -->I$. If $f(b)>f(a)\mathrm{\forall <!--\; \forall \; -->}b>a$, the function is said to be strictly increasing.
Conversely, A function f(x) decreases on an interval I if $f(b)\le f(a)\mathrm{\forall <!--\; \forall \; -->}b>a$, where $a,b\in <!--\; \in \; -->I$. If $f(b)<f(a)\mathrm{\forall <!--\; \forall \; -->}b>a$, the function is said to be strictly decreasing.
Then how would a horizontal line be described? If $f(x_2)=f(x_1)\mathrm{\forall <!--\; \forall \; -->}{x}_2>x_1$ does that mean that a horizontal line meets the definition of an increasing function and a decreasing function?

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Find the direction cosines of the a vector in the plane of $\stackrel{\to <!--\; \to \; -->}{b}=<2,1,3>$ and $\stackrel{\to <!--\; \to \; -->}{c}=<-<!--\; -\; -->1,2,1>$ and perpendicular to $\stackrel{\to <!--\; \to \; -->}{c}$
Let the vector $\stackrel{\to <!--\; \to \; -->}{a}=⟨<!--\; ⟨\; -->x,y,z⟩<!--\; ⟩\; -->$
Since it is perpendicular to $\stackrel{\to <!--\; \to \; -->}{c}$
$-<!--\; -\; -->x+2y+z=0$
And
$\left|\begin{array}{ccc}x& y& z\\ 2& 1& 3\\ -<!--\; -\; -->1& 2& 1\end{array}\right|$
$⟹<!--\; ⟹\; -->-<!--\; -\; -->5x-<!--\; -\; -->7y+5z=0$
since a, b, c are coplanar
So y=0
Then x=y
So the answer should be $⟨\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}⟩$ , but that is not correct.
What is wrong with my solution?

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Let ${\displaystyle y=f\left(x\right)}$ be a function D=(-12,13) and range ${\displaystyle R=(-\mathrm{\infty },\mathrm{\infty }).}$ Find the domain D and range R for the following functions and enter your answers using interval notation. Keep in mind order of operations. Be sure your intervals are in the correct order, and enter exact answers only (no approximations).
a) ${\displaystyle y=f\left(\frac{1}{2}x\right)}$
b) ${\displaystyle y=5f\left(2x\right)-2}$

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) You have cashews containing 170 calories, 115 mg of sodium and 14mg of fat. Then you have peanuts with 150 calories, 110mg of sodium and 12mg of fat. Finally, you have walnuts with 160 calories,270 mg of sodium and 11mg of fat. You want to produce 10 batches each of which has 220 calories, 188 mg of sodium and 17.4 mg of fat. How many of each do you need for the mixture?

Find the limits
$\underset{x\to 0}{lim}\sec [e^x+\pi \tan (\frac{\pi }{4\sec x})-1]$

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Weaknesses of Wald confidence interval for binomial distribution
A survey samples 1000 people, among which 500 say they will vote for A, 400 for B e 100 for C. Calculate a confidence interval for the proportion of people that will vote for A. What are the major weaknesses of the confidence interval you just calculated?
Let ${\displaystyle A_n\sim \text{Binom}(n,p)}$. By the CLT,
${\displaystyle \sqrt{n}({\bar{A}}_n-p)x\to \left\{d\right\}\mathcal{N}(0,p(1-p))}$
hence ${\displaystyle \frac{\sqrt{n}}{\sqrt{p(1-p)}}({\bar{A}}_n-p)x\to \left\{d\right\}\mathcal{N}(0,1)}$
Since p is unknown, we approximate it with ${\displaystyle {\bar{A}}_n}$. It follows that, at ${\displaystyle 1-\alpha }$ confidence,
${\displaystyle p\in {\bar{A}}_n\pm z_{1-\frac{\alpha }{2}}\sqrt{\frac{{\bar{A}}_n(1-{\bar{A}}_n)}{n}}}$
Since this confidence interval relies on the CLT, it gives poor results when n is small or p is very close to 0 or 1. But this is not the case here, as ${\displaystyle n=1000}$ and ${\displaystyle {\bar{A}}_n=0.5}$, so I fail to see what weaknesses the question is talking about?

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differential equations of second order
How may I solve this differential equations:
$$y^{″}+4y=12x^2-<!--\; -\; -->16x\cos <!--\; \; -->(2x)?$$

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