0
votes
1answer
41 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
41 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
70 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
79 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
74 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
22 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
29 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
32 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
60 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
110 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 ...
3
votes
4answers
237 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 ...
5
votes
3answers
89 views

prove that $a^b\ge{b}^a$ where $a\le{b}$.

prove that $a^b\ge{b}^a$ for all $a,b\ge3$. given that $a\le{b}$. I was trying to solve the question by graph. Can anyone help me please?
1
vote
1answer
65 views

Ratio of 2 Gammas, approximation with power

Find all value of $\alpha$ such that $\lim\limits_{x\rightarrow +\infty}\left(\frac{\Gamma(x+\alpha)}{\Gamma(x)}-x^{\alpha}\right)=0$. (note: $\alpha$ is a constant with respect to $x$) By ...
0
votes
1answer
48 views

How do I prove the following inequality $\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}$?

I would appreciate some help proving the inequality $$\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}.$$ Thanks in advance!
3
votes
2answers
76 views

there exist some real $a >0$ such that $\tan{a} = a$

How can i prove that there exist some real $a >0$ such that $\tan{a} = a$ ? I tried compute $$\lim_{x\to\frac{\pi}{2}^{+}}\tan x=\lim_{x\to\frac{\pi}{2}^{+}}\frac{\sin x}{\cos x}$$ We have the ...
0
votes
0answers
19 views

Application of mean value theorem to function $x \to (x-y)^{a-1-d/2}$

How can I show the following inequality by mean value theorem, for a constant $C>0$ $2|(x+h-y)^{a-1-d/2} - (x-y)^{a-1-d/2}|^p \leq C (x-y)^{(a-2-d/2)p}h^p$ Proof: Let $f(b) =(x+h-y)^{a-1-d/2}$, ...
1
vote
0answers
24 views

$\frac {1 } {10 }(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2)^2+(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1)^2) \le (y_1-x_1)^2+(y_2-x_2)^2$?

Is it true that: $$\frac {1 } {10 }\left(\left(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2\right)^2+\left(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1\right)^2\right) \le (y_1-x_1)^2+(y_2-x_2)^2$$ I think I should ...
2
votes
3answers
74 views

Proving $\frac2\pi x \le \sin x \le x$ for $x\in [0,\frac {\pi} 2]$

Prove $\frac2\pi x \le \sin x \le x$ for $x\in [0,\frac {\pi} 2]$. I tried to do this in two ways, I'm not sure about CMVT and I have a problem with the other way. Using Cauchy's MVT: RHS: ...
6
votes
2answers
85 views

Advice for self-studying Inequalities and Calculus

I'm interested in self-studying the following books over the next year or so: Spivak's Calculus (I'm already in Ch. 5 and it is very slow going) The Cauchy-Schwarz Master Class by J. Michael Steele ...
4
votes
3answers
86 views

Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$

Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0<a\le a_n\le b <\infty.$ Prove: $\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 ...
2
votes
2answers
64 views

Prove that $(1+x)\ln(1+x) >\arctan(x)$

My solution is that slope of $\ln(1+x)$ is greater than $\arctan(x)/(1+x)$ & at $x=0$ both of these are equal and hence inequality proved. What i am looking for is the restrictions on $x$ in which ...
6
votes
6answers
138 views

Show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$

This is a pretty straightforward question. I want to show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$. One way would be this. WLOG, suppose $x \leq y$. Then: $(1+\frac{x}{y})^a ...
6
votes
4answers
152 views

How can we prove $\int_1^\pi x \cos(\frac1{x}) dx<4$ by hand?

Is there any way we can prove this definite integral inequality by hand: $$ \int_{1}^{\pi}x\cos\left(1 \over x\right)\,{\rm d}x < 4 $$ I don't where to start even, please help. That ...
1
vote
2answers
50 views

Differential equation with sec

With $(a)$ I got that $-y^2 dx = \sec^2x\ dy$, but it makes no sense. Hence, no Idea how to handle $(b)$.
1
vote
4answers
65 views

Is it true that $a^p b^q \leq a+b$ with $p + q = 1$?

Let $a,b \geq 0$ and $0<p,q < 1$ s.t. $p + q = 1$. Is it true that $a^p b^q \leq a+b$?
2
votes
1answer
44 views

(Elementary) Trigonometric inequality

Any idea for proving the following inequality: $5+8\cos x+4 \cos 2x+ \cos3x\geq 0$ for all real x? I've tried trigonometric identities to make squares appear, and other tricks; but nothing has worked ...
1
vote
1answer
25 views

differential inequality of continuous functions

Let $u:[0,+\infty)\to (0,+\infty)$ be a continuous function such that $\int_0^\infty u(x)dx<\infty$. Suppose there exist $a,b>0$ such that $\frac{du}{dx}\leq u(a+bu)$. Prove that ...
1
vote
4answers
125 views

Two ways to show that $\sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$

Show that: $\large \sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ on: $0<x<\frac {\pi}2$ I tried to solve it in two ways and got a little stuck: One way is to use Cauchy's MVT, define $f,g$ ...
0
votes
2answers
62 views

How would this problem need the Mean Value Theorem?

I'm asked to square the inequality and use the Mean Value Theorem to prove that $$\sqrt{1+x} < 1 + \frac{x}{2}$$ for $x>0$. Unfortunately, I don't really understand why I would need such a ...
0
votes
1answer
43 views

Help with solving a triangle inequality

I'm trying to solve the following inequalities using the triangle inequality but I'm not sure on how. I got the following two inequalities: $xf'(x)-f(x)\le \frac {x^2} 2 A$ $f'(x)-(xf'(x)-f(x))\le ...
1
vote
1answer
57 views

How find the minimum of $a$ ,if $f(x)=-\frac{\ln{x}}{x}+e^{ax-1}-a,x>0$

Question: let $$f(x)=-\dfrac{\ln{x}}{x}+e^{ax-1}-a,x>0$$ if such $$f(x)_{min}=0,\forall x>0$$ **Question: Find the $a$ minimum of the value. My idea: this problem equivalent to ...
2
votes
1answer
76 views

Tighter logarithmic inequality

There is a well-known lower bound for $$ x\log{1+x\over x}\geq {x\over1+x} $$ for $x\geq0$. I know a tighter lower bound on the same domain $$ x\log{1+x\over x}\geq{2x\over1+2x}\geq {x\over1+x}. $$ It ...
1
vote
1answer
31 views

Why does showing a function that is comprised of an inequality has an extremum prove the inequality?

This is related to prove: $x^\alpha - \alpha x \le 1-\alpha$ and Proving $x^\alpha-\alpha x \le 1- \alpha $. You can see in the answers, they turn the inequality to a function, take the derivative ...
1
vote
2answers
75 views

An exponential inequallity $2^x > 1-x$

$$ 2^x > 1-x, \\ e^{x\ln2} > 1-x, \\ (1-x)e^{-x\ln2} < 1, \\ (1-x)e^{(1-x)\ln2} < e^{\ln 2} = 2^1, \\ (1-x)^2e^{\ln2} < e^{\ln 2}, \\ (1-x)^2 < 1, \\ x^2-2x+1-1 < 0, \\ x(x-2) ...
2
votes
0answers
55 views

How to prove or disprove this statement?

For $n\geq1$, $$\int_{0}^{\pi/2}(\theta \sin\theta)^{n+1}d\theta>\int_{0}^{\pi/2}(\theta \sin\theta)^{n}d\theta$$ . It is hard to find $\int_{0}^{\pi/2}(\theta \sin\theta)^{n}d\theta$ so I have no ...
0
votes
2answers
75 views

What's the proper name of this theory?

Firstly, I'm not sure if this is valid or not, but so far it's been giving me the desired result, so if this theory is invalid, please let me know the exceptions. To get the maximum result from the ...
5
votes
2answers
193 views

How prove that there exists $m$ such that $|f(m)|\le \dfrac{\sqrt{b^2-4c}}{2}$

Let the function $$f(x)=x^2+bx+c,\qquad b^2-4c>0$$ Assume that $x_{1},x_{2}$ are the roots of $f(x)$ and $|x_{1}-x_{2}|\ge 1$. Show that: There exists an integer $m$, such that $$|f(m)|\le ...
1
vote
1answer
31 views

The Minimum of a Sum of Continuous functions

Suppose you have two continuous functions: $ f, g : [0, \infty) \to \mathbb{R}$ and you already know that a lower bound for $(f+g)$ is $0$---that is, $(f+g)(x)>0, \forall \; x \in [0, \infty)$. ...
1
vote
1answer
46 views

Mathematical Proof (Apostol)

If $x > 0$, prove that there is a positive integer $n$ such that $\frac{1}{n} < x$ byy either contradiction or contrapositives. My attempts By contrapositives: Givens by contrapositive method ...
1
vote
3answers
40 views

Inequality for quartic polynomial depending on a parameter

Let $f(x) = \frac 14 x^4 - \frac \alpha2 x^2 - (\alpha-1) x - \frac \alpha 2 + \frac 3 4 $. I want to show that there exists an $\alpha>1$ such that $f(x)\geq 0 $ for $x\leq 0$. Even more, it ...
0
votes
2answers
52 views

$(x+y)^{p/2} \leq x^{p/2} + y^{p/2} (1<p<2)?$

Let $x, y \in (0, \infty)$ and $1<p<2.$ My Question: Is it true that, $$(x+y)^{p/2} \leq x^{p/2} + y^{p/2} ?$$
6
votes
1answer
57 views

Differential inequality implies inequality for points at distance pi.

Given a function $f$ with $f+f''\ge 0$, show that $f(x)+f(x+\pi) \ge 0$ for all $x$. Note that for sine and cosine both inequalities become equations. It seems reasonable to look at $f+f''=g$, but ...
1
vote
1answer
23 views

Basic Inequality

I have that $|L_j|<(\Delta t) A_1+ hA_2$ I need it in the form $|L_j|<(\Delta t+h)A_L$ where $A_L=A_L(A_1,A_2)$ and $A_1, A_2,A_L$ are all constants. I have thought maybe combing the facts ...
1
vote
1answer
76 views

The inequality about recurrence sequence

Sequence $(x_n)$ is difined $x_1=\frac {1}{100}, x_n=-{x_{n-1}}^2+2x_{n-1}, n\ge2$ Prove that $$\sum_{n=1}^\infty [(x_{n+1}-x_n)^2+(x_{n+1}-x_n)(x_{n+2}-x_{n+1})]\lt \frac {1}{3} $$ I found relation ...
0
votes
2answers
17 views

$\big| |z|-|w|\big| \leq |z-w| \implies \big|c_{1} |z|- |w|c_{2}\big| \leq |c_{1}z- c_{2} w| ? $ ($z, w \in \mathbb C, C_{1}, C_{2} >0$)

By triangle inequality, we get, $$\big| |z|-|w|\big| \leq |z-w|; (z, w\in \mathbb C.)$$ Take any $C_{1}, C_{2} > 0$ and fix it. My Question is: Can we expect: $$\big|C_{1} |z|- |w|C_{2}\big| ...
0
votes
2answers
15 views

$|z_{1}- z_{2}| \leq |w_{1}- w_{2}| \implies |c_{1}z_{1}- c_{2}z_{2}| \leq |c_{1}w_{1}- c_{2} w_{2}|$?

Let $z_{1}, z_{2}, w_{1}, w_{2} \in \mathbb C$ with $|z_{1}- z_{2}| \leq |w_{1}- w_{2}|.$ Fix $c_{1}, c_{2}\in (0, \infty).$ My Question is: Can we expect, $|c_{1}z_{1}- c_{2}z_{2}| \leq ...
0
votes
0answers
24 views

Proof metric space with distance function

Thats the first time i have to do such an proof but don't know how, never seen or done this before. Especially (iii). Let $X$ be the Set of all complex sequences. $$ d((a_n),(b_n)) := ...
-2
votes
2answers
70 views

show $\frac{x-1}{x}<\log x<x-1$ using Lagrange's mean value theorem [closed]

Using Lagrange's mean value theorem, prove that $$\frac{(x-1)}{x} < \log x < x-1$$ for $x>1$. What should f(x) be? I dont know how to go about such question.