18 votes
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How to show analytically that $x^4-4x^3-2x^2+16x+24=0$ has no solutions in $\mathbb{R}$?

$$f(x)=x^4-4x^3-2x^2+16x+24=(x^2-2x-4)^2+2x^2+8>0.\tag{1}$$ My motivation: the terms $-4x^3$ and $16x$ are annoying since they can be either positive or negative, so I wanted to put them in a ...
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  • 8,591
13 votes
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When $f(z) = 1/z$ and $g(z) = 1-z$, why is $f \circ g \circ f \circ g \circ f \circ g (z) = g \circ f \circ g \circ f \circ g \circ f (z) = z$?

The functions $f(z) = 1/z$ and $g(z) = 1 - z$ are both linear fractional transformations; that is, they have the form $\frac{az + b}{cz + d}$ for suitable complex numbers $a, b, c, d$ such that $ad - ...
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8 votes
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Does a function $f(x)$ exist such that $f(x+1)-f(x)=\frac{1}{x}$. And if so what is it.

Yes. Let $L(z)$ denote the logarithm of the gamma function. Then $L(x + 1) - L(x) = \log(x)$. So the derivative $L'(x)$ (i.e. $\Gamma'(x) / \Gamma(x)$) satisfies the desired functional equation.
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  • 25.7k
6 votes

Evaluating the value of first derivative at $x=1$ for a polynomial $f$ satisfying $f(x)+f'(x)+f''(x)=x^5+64$

$$f(x) + f'(x) + f''(x) = x^5 + 64$$ Take the derivative. $$f'(x) + f''(x) + f'''(x) = 5x^4$$ Subtract the two equations: $$f(x) - f'''(x) = x^5 - 5x^4 + 64$$ Differentiate some more. $$f'(x) - f''''(...
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  • 9,503
6 votes

Evaluating the value of first derivative at $x=1$ for a polynomial $f$ satisfying $f(x)+f'(x)+f''(x)=x^5+64$

If you were not told that $f(x)$ is a polynomial, then you just need to solve the differential equation $$f(x)+f'(x)+f''(x)=x^5+64$$ With the usual method, the solution is $$f(x)=e^{-x/2} \left(c_1 \...
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6 votes

Let $f(x),f'(x),f''(x)$ be all positive for all $x\in[0,7]$. If $f^{-1}(x)$ exists then $f^{-1}(5)+4f^{-1}(6)-5f^{-1}(\frac{29}5)$

The function $f$ is strictly monotonously increasing $(f'>0)$ and strictly convex $(f''>0)$. If $f^{-1}$ exists, it is strictly monotonously increasing and strictly concave. Hence, \begin{align}\...
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  • 1,942
6 votes
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Can we compute this integral $\int_0^{2\pi} \frac{1}{5+3 \cos x} dx$ using an antiderivative defined on $\mathbb R$?

So with that Weierstrass substitution we have $$\newcommand{\II}{\mathcal{I}} \newcommand{\dd}{\mathrm{d}} \mathcal{I} := \int_0^{2\pi} \frac{1}{5+ 3 \cos x} \, \dd x = \int_?^? \frac{1}{5+3(1-t^2)/(1+...
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  • 31.8k
5 votes
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Prove that $1/x^2$ is not uniformly continuous on $(0,\infty)$ using $\varepsilon$-$\delta$ arguments

Claim: $f:(0,\infty)\to \Bbb{R}$ defined by $f(x)=\frac{1}{x^2}$ is not uniformly continuous. Preliminary: $f$ is uniformly continuous if, for every $\epsilon>0$, there exists a $\delta>0$ such ...
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  • 8,918
5 votes

Find the number of solutions to the equation $\sin\left(\frac{\pi\sqrt x}4\right)+\cos\left(\frac{\pi\sqrt {2-x}}4\right)=\sqrt 2$

From the comments, the domain is $[0,2]$. Let $f(x)=\sin\left(\frac{\pi\sqrt x}4\right)+\cos\left(\frac{\pi\sqrt {2-x}}4\right)-\sqrt 2$ $f'(x)=\frac1{2\sqrt x}\frac\pi4\cos\left(\frac{\pi\sqrt x}4\...
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  • 5,765
5 votes
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Why is Desmos not showing $\ln(y)-\ln(y-1)$ as the same as $\ln(y/(y-1))$?

The reason is simple: $f(x) := \ln(x) - \ln(x-1)$ is always positive, since $\ln(x-1) < \ln(x)$. Note that we cannot have $x \le 1$ for this equation to be well-defined. $g(x) := \ln (x/(x-1))$ is ...
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  • 31.8k
4 votes
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Centroid coordinate for region

Yes, the answer is $\frac7{20}$. You should have computed$$\frac{\displaystyle\int_0^1\int_{2x}^{3-x^2}\color{red}x\,\mathrm dy\,\mathrm dx}{\displaystyle\int_0^1\int_{2x}^{3-x^2}1\,\mathrm dy\,\...
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4 votes
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Given graph of the function $f(x)=x^3+ax^2+bx+c$ What is the $x$ value at the local minimum point?

From the shape of the graph you have two roots, call them $p$ for the single on the left and $q$ for the double root on the right, so the equation can be factorised as $(x-p)(x-q)^2=0$. This results ...
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  • 7,503
4 votes
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Evaluating the value of first derivative at $x=1$ for a polynomial $f$ satisfying $f(x)+f'(x)+f''(x)=x^5+64$

(This answer originates from this page given by @DatBoi in the comments under the question.) $$ \lim_{x\rightarrow 1}{f(x)\over x-1}=f'(1) $$ (and $f(1)=0$) $$ f(x)+f'(x)+f''(x)=x^5+64\\ f'(x)+f''(x)+...
4 votes
Accepted

What is the continuous version of "diagonal matrix"?

The generalization you are looking for is the multiplication operator (examples in link). The spectral theorem takes the following form in this case: Link.
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  • 1,405
4 votes
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Work done by a field along a circle

You can't use Green's theorem because you don't have continuous partial derivatives. Let $G = (\frac {-y}{(x-1)^2 + y^2},\frac {x-1}{(x-1)^2+y^2})$ and $H = (\frac {-y}{(x+1)^2 + y^2},\frac {x+1}{(x+1)...
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  • 6,785
4 votes
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How can I invert a complicated bivariate polynomial?

In general, it is impossible to write down a formula for the inverse function of a polynomial with degree $\ge 5$. This result is known as Abel's impossibility theorem. For example, if you want to ...
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4 votes
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Proving that the map $f:\mathbb R \to \text{Seq}(\mathbb Q)/\sim$ is surjective

If $(a_n)_{n\in\Bbb N}$ be a Cauchy sequence of rational numbers, and let $a=\lim_{n\in\infty}a_n$; it exists, since any Cauchy sequence of real numbers converges. Then $f(a)$ is the equivalence class ...
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4 votes
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Minimum of a function defined by a sum

I think I have found an answer for the even case (when $n$ is even). Indeed, let $n=2p$ and since $\varphi(x) = -\varphi(2p-x)$ it is enough to study the following case: $x\in(2l-1, 2l)$ for $l\in\{1,...
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  • 5,288
4 votes

At what point do two lines extending from the end of a third equal each other?

We put the whole configuration on a Cartesian OXY Coordinate System, with the common center as the origin. We know, any point on a circle with center as origin and radius $r$ can be described ...
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4 votes
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Finding the maximum value of $f(x)=\frac{|x|-2-x^2}{|x|+1},x\in\mathrm R$

Let $t=|x|+1\geq 1$, then $$\frac{|x|-2-x^2}{|x|+1}=\frac{t-1-2-(t-1)^2}t=\frac{-t^2+3t-4}t=3-\left(t+\frac4t\right).$$ By AM-GM, $t+\frac4t\geq 2\sqrt{t\cdot \frac4t}=4$, with equality iff $t=2$, ...
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  • 8,591
4 votes
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Find the number of bijective function such that $f(3) \geq f(9) \geq \ldots \geq f(99)$

Let $B = \{2,4,6,8,\ldots,100\}$, $n=|B|$, $k = |\{3,9,15, ... 99\}|$. Choose a subset of $B$ of size $k$. Assign values from that subset to $f(3), f(9), f(15), ... f(99)$, such that condition $f(3) ...
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4 votes

Help understanding example of not a function

A function is defined as having exactly one output for each input in the domain. The expression $\pm \sqrt{\frac 13x}$ does not provide a unique output for every input $x$, so it does not represent a ...
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3 votes

Finding domains of functions such as $y = \sqrt{25-5^x}$, $y = e^{\arcsin x}$, $y = \cos^{-1}\left(e^{3x}\right)$

For the first example, we want to find all $x$ such that $$25-5^x \geq 0 \implies 25 \geq 5^x.$$ You know that $5^2 = 25$. Now, because the exponential function $5^x$ increases as $x$ increases, you ...
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  • 1,089
3 votes
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check the set of point of Continuity of the function

Hint Any sequence $\{x_n\}$ is composed of a subsequence of rational points $\{x_n^r\}$ and a sequence of irrationals $\{x_n^i\}$. And to prove that such a sequence converges, it is sufficient to ...
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3 votes

How do I find the exponential equation from just its graph?

Plug the given values into the equation. If you plug in the $x$ and $y$ values at $x = 0$, you would get this equation: $$b e^0 + 20 = 22.5$$ $$\implies b + 20 = 22.5 \implies b = 2.5$$ Using that $b$ ...
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3 votes

Professional programmer struggling to learn Calculus

I’ll walk you through a specific example and hopefully that will clarify your problem. Notation Let’s define a simple polynomial $f(x)=x^2$ where $x$$\in$$R$ (where $R$ is the set of all real numbers)....
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  • 1,173
3 votes

Is it possible to solve the equation $x - 1 = x^{-y}$ explicitly?

You can find $y$ in terms of $x$ by taking (natural) logarithms: $$ \log(x-1)=-y\log x $$ and therefore $$ y=-\frac{\log(x-1)}{\log x} $$ which is defined for $x>1$. The function is decreasing: ...
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  • 230k
3 votes

Given graph of the function $f(x)=x^3+ax^2+bx+c$ What is the $x$ value at the local minimum point?

Let $s$ and $t$ denote the $x$-intercepts. Then $$ f(x)=(x-s)(x-t)^2 $$ Since we know $f(0)=4$ that gives $-st^2=4$ so $s=-\frac{4}{t^2}$ Next, we have $$f^\prime(x)=(x-t)^2+2(x-s)(x-t)$$ Since $f^\...
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3 votes

Can we compute this integral $\int_0^{2\pi} \frac{1}{5+3 \cos x} dx$ using an antiderivative defined on $\mathbb R$?

Your first antiderivative is "correct", that isn't (quite) the problem. When I say "correct" rather than correct, the point is that to speak accurately we should always talk about ...
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  • 77.6k

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