# Tagged Questions

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### $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l}$

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l}$ for every $m, n, l >0$.
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### Prove $\sin(x)< x$ when $x>0$ using LMVT

According to Lagrange's Mean Value Theorem (LMVT), if a function $f(x)$ is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$, then there exists some constant $c$ such that ...
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### Finding the range of $x$ in ${c_1}^x+\sqrt{\frac{\log(c_2)x}{c_3}}+\frac{\log(c_2)x}{c_4}\le c_5,$

Is there any way to find the range of $x$ that satisfies the following inequality: $${c_1}^x+\sqrt{\frac{\log(c_2)x}{c_3}}+\frac{\log(c_2)x}{c_4}\le c_5,$$ for $x>1$, $0<c_1<1$, ...
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### Proof of inequality $\frac{2-a}{2+a}<e^{-a}$

How can I prove that $$\frac{2-a}{2+a}<e^{-a}$$ for all $a \geq 0$ ? For $a \geq 2$ it is clear, but how can it be shown for $0<a<2$ ?
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### Showing that $x+ cos x - 1 > 0$ for all $x > 0$

I got this problem: Show that for all $0<x$, $0<x+cos x - 1$ I tried to show it several times but none worked. I showed that $lim_{x\to\infty} (x+cos x - 1) = \infty$ by using the Squeeze ...
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### If$x^2 + y^2 + Ax + By + C = 0 .$ Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.

Also find the center and radius of the circle Here's my solution, I'm not sure if it's correct or not (specifically the conditions on $A$, $B$ and $C$. I feel that my conditioning is invalid and that ...
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### Show that the solution of an initial value problem is always less than a given constant

My try is that $$\frac{dy}{dt} =(y-3)e^{\cos ty}$$ $$\frac{dy}{y-3}= e^{\cos ty}dt$$ $$\ln (y-3)=-\frac{e^{\cos ty}}{\sin ty} +c$$ my steps is correct or I made mistakes ? please help to solve ...
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### $x^2+y^2+z^2 +3 \geq 2(xy+xz+yz)$, for $xyz=1$ [closed]

$x,y,z > 0$ such that $xyz=1$ can you prove that $x^2+y^2+z^2 +3 \geq 2(xy+xz+yz)$ without lagrange multiplier Thanks
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### How to show that $a+b> \sqrt{a^2+b^2-ab}, \qquad a, b >0$

How do you show that $$a+b> \sqrt{a^2+b^2-ab}, \qquad a, b >0$$ I could write $\sqrt{a^2+b^2-ab}=\sqrt{(a+b)^2-3ab}$, but this seems to lead nowhere.
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### verifying extrema found by Lagrange multipliers

This question was inspired by reading this problem: Prove the inequality $\frac 1a + \frac 1b +\frac 1c \ge \frac{a^3+b^3+c^3}{3} +\frac 74$ Suppose I have a function $f(x,y,z)$ with continuous ...
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### Conditional extreme value of a function

Let $x,y,z$ be the positive real numbers, if $x^2+y^2+z^2=1$, then how can we find the minimal value of this function $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.
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### Solving the logarithimic inequality $\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$

I tried solving the logarithmic inequality: $$\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$$ several times but keeping getting wrong answers.
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### inequality funny question

I'm not sure what they want here: solve the inequality in realtion to $x$ for various values of $a$ : $\frac{(a+2)x}{a-1} - \frac{2}{3} < 2x-1$
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### Show that for all $(\tau, \xi) \in \mathbb R^{n+1}$ we have $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$

Show that, for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$ This is the exercise 7.4 in the book by Francois Treves. It is just a fundamental ...
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### How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
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### Arc Length of a Curve

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, how can you prove (not in the geometric way): $$\sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx$$
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### Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
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### solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x}$$
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### $|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \$ [closed]

Show that for every arbitrary complex number a,b and c we have $$|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \$$ Thanks.
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### Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
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### How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
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### Use mean value theorem on $f(x) = x^{1/5}$, to show that $2< \sqrt[5]{33}<2.0125$

The problem specifically aks us to use mean value theorem on the interval $[32, 33]$ It has always puzzled me that mean value theorem can be used to prove Inequalities. Can anyone show how mean ...
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### Show that $\int_{\pi/4}^{\pi/2} \frac{\sin x}{x}\,dx\leq \frac{\sqrt{2}}{2}$

Show that $$\int_{\pi/4}^{\pi/2} \dfrac{\sin x}{x}\,dx\leq \dfrac{\sqrt{2}}{2}$$ Any Ideas, how to start ?!
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### Inequality containing finite sum.

For what value of k the following inequality holds? $\sum_{i=1}^{n}a_{i}^3<k|\sqrt{\sum_{i=1}^{n}a_{i}}|$ I don't have any idea to solve this.
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### Solve this with CBS

How can you see the mínimum value of $1/x + 4/y + 9/z$ with x+y+z=1 using the CBS inquality? I have seen a proof of that that use trigonometric substitutions, but i don´t see as one-step the ...
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### Typo in Spivak's explanation of limits in Calculus?

Here's what he says (including the preceding paragraph): "To show in general that f [(where f(x)=1/x)] approaches 1/a near a for any a we proceed in basically the same way, except that, again, we ...
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### Is $-|x|\le\sin x\le|x|$ for all $x$ true?

I have seen in Thomas' Calculus that says to prove $\lim_{x\rightarrow0}\sin x=0$, use the Sandwich Theorem and the inequality $-|x|\le\sin x\le|x|$ for all $x$. My question is how could the ...
Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
### What will be the range of $f(x)= \frac{12}{\sqrt{(15-2x-x^2)}}$
Here's my try: Since the denominator involves a square root so I solved the following inequality: $15-2x-x^2>0$ which gives a solution set of $x=(-5,3)$. This is the domain of $f(x)$. However since ...