1
vote
2answers
42 views

$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 $m, n, l >0$.
0
votes
1answer
64 views

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 ...
0
votes
0answers
14 views

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 $x>1$, $0<c_1<1$, ...
-2
votes
4answers
93 views

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 $0<a<2$ ?
2
votes
3answers
45 views

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 ...
1
vote
1answer
35 views

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 ...
4
votes
3answers
59 views

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 ...
0
votes
1answer
60 views

$x^2+y^2+z^2 +3 \geq 2(xy+xz+yz)$, 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
1
vote
4answers
86 views

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.
2
votes
0answers
61 views

An entropy inequality

Let $f:[0,2\pi]\to \mathbb{R}$ be a smooth, positive function such that $f(0)=f(2\pi)$, and $\int_0^{2\pi}fd\theta=2\pi.$ Is it true that $$2\int_0^{2\pi}f\ln fd\theta- 2\int_0^{2\pi}\ln ...
0
votes
2answers
64 views

Solve $|x-2| \leq 2|x|$

This is an in-class example we were given in calculus class, I am having some difficulty understanding one of the instructor's steps. The following is my attempt of the question: Since this is an ...
1
vote
1answer
35 views

Inequalities with function $e^{x^2+e^{x^2}}$

Let $f(x)=e^{x^2+e^{x^2}}$ for $x\in\mathbb{R}$. How to prove that for any $a,b>0$, $a\neq b$ the following inequalites hold $$(b-a)f(\frac{a+b}{2}) < \int_a^b f(x)\ dx < ...
3
votes
1answer
54 views

Does such a function exist always?

Suppose that $f(x)$ is some smooth function on $[0,1]$ with $f(x) \geq c > 0$. Can we always find a function $g(x)$ smooth satisfying $g'(x) \not= 0$ for all $x \in [0,1]$ and $f'(x)g'(x) + ...
1
vote
1answer
45 views

Proving inequality $\frac{1}{2}e^x\left(2+e^x\right) > \left(1+e^x\right)\ln(1+e^x)$

Do you have any ideas on how to show the following inequality? $$\frac{1}{2}e^x\left(2+e^x\right) > \left(1+e^x\right)\ln(1+e^x)$$ It's not about the convexity of any of those functions. ...
0
votes
1answer
79 views

How to find numbers $k$ such that $kx - \ln(ex + 1-x) $ is positive on $(0,1]$?

I want to find a condition on $k$ such that $g(x)= kx - \ln(ex + 1-x) > 0$, $x\in [0,1] $. At zero the function is zero. So, to find a condition on $k$ I use $g'(x) > 0$ i.e. $$ k > ...
1
vote
2answers
95 views

In the proof of $\lim_{x\rightarrow 1}(2x-1) = 1$, why do we choose $\delta=\epsilon/2$?

First of all, I didn't understand why mathematicians call it a proof?is it really a proof?What are we trying to prove?It's kind of like we're putting some numbers and the equality just works. Consider ...
3
votes
1answer
80 views

Help with problem 1-5 from Calculus by Spivak (multiplying two inequalities together)

This question involves problem 5(viii) from chapter 1 of Spivak's Calculus, third edition. I'm a layman trying to teach myself some more advanced mathematics, and although I've been making slow ...
0
votes
2answers
35 views

Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
2
votes
2answers
78 views

How to prove $n! > n^a$ for all $a\in \mathbb{R}$ (for sufficiently large $n$)?

I've encountered a proof which claims $n! > n^2$ for sufficiently large $n$. I tried using induction to prove it for an arbitrary $a$: $n! > n^a$. Lets assume the claim is true for $n$: $n! ...
1
vote
0answers
79 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
1
vote
1answer
65 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
1
vote
1answer
61 views

Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the ...
2
votes
2answers
29 views

Taking root from absolute expression

Why is the following true? (Where all terms are positive) $$|x-y| < \epsilon^2 \implies |\sqrt x - \sqrt y| < \epsilon$$
1
vote
2answers
36 views

a proof of constants are null from a given inequality

Problem: given constants $a,b\text{ and }c$, and a variable $x$, assume that for all $x\in\mathbb{R}$ holds that $|ax^2+bx+c|\le|x|^3$, then proof that $a=b=c=0$ My try: substitute $x=0$ into the ...
0
votes
1answer
55 views

Show That $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ And $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ Are Orthogonal Trajectories

Show that the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the hyperbola $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ are orthogonal trajectories if $A^2< a^2$ and $a^2-b^2 = A^2+B^2$. What I've ...
6
votes
5answers
205 views

Alternate Proof for $e^x \ge x+1$

This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it: Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) ...
2
votes
1answer
43 views

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 ...
1
vote
2answers
91 views

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 $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.
1
vote
4answers
130 views

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.
0
votes
1answer
65 views

inequality funny question

I'm not sure what they want here: solve the inequality in realtion to $x$ for various values of $a$ : $\frac{(a+2)x}{a-1} - \frac{2}{3} < 2x-1$
1
vote
2answers
48 views

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 ...
9
votes
11answers
416 views

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
5
votes
4answers
367 views

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): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
10
votes
4answers
327 views

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 ...
2
votes
2answers
122 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
2
votes
0answers
50 views

$|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.
0
votes
0answers
22 views

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 ...
0
votes
1answer
43 views

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 ...
0
votes
1answer
46 views

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 ...
0
votes
1answer
73 views

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 ?!
5
votes
4answers
143 views

How find the maximum of the $x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$

Let $$0\le x_{i}\le i,\, i=1,2,3$$ be real numbers. Find the maximum of the expression $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$$ My idea: I guess $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}\le ...
2
votes
1answer
84 views

Homework on basic inequalities.

Let $a_j$ be a sequence of positive reals. Show that (a) $\left(\sum_{j=1}^\infty a_j\right)^\theta \le \sum_{j=1}^\infty a_j^\theta$ for any $0\le\theta\le1$. (b) $\sum_{j=1}^\infty a_j^\theta \le ...
4
votes
2answers
89 views

Binomial expansion inequality

In a paper I am reading, there is a step that seems to come from the following inequality: $$(1+x)^\alpha \le 1+2^\alpha x,$$ where $0<x<1$. (Also, $3\le \alpha \le 9/2$ in the context of the ...
0
votes
1answer
25 views

How to prove this elementary “ interpolation” inequality?

Suppose $2<p<\infty$ and $0<\theta<1$. Let $n\geq 1$ be an integer. Assume that $$ \frac{1}{p}=\frac{1-\theta}{2^n}+\frac{\theta}{2^{n+1}}. $$ How to prove the following inequality $$ ...
3
votes
1answer
31 views

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.
1
vote
1answer
35 views

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 ...
3
votes
1answer
70 views

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 ...
1
vote
3answers
113 views

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 ...
2
votes
4answers
247 views

inequality method of solution

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 ...
1
vote
2answers
32 views

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 ...