0
votes
2answers
62 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
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 ...
3
votes
1answer
55 views

Range of $f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}$ for a specified domain

We are asked to find the range of the function $$f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}, \;\;\text{for}\;0\le x\le2\pi$$ I tried to find the range of each basic function of cos and sin then ...
2
votes
1answer
59 views

Figuring out when $f(x) = \sin(x^2)$ is increasing and decreasing

Regarding the function $f(x) = \sin(x^2)$, I'm supposed to figure out when it is increasing/decreasing. So far, I've found the derivative to be $f'(x) = 2x\cos(x^2)$. So long as I can solve the ...
0
votes
0answers
36 views

Want to find a function $f:\mathbb{R} \to \mathbb{R}$ that satisfies these inequalities

I want to find a function $f \in C^\infty([0,T])$ such that $$0 < L \leq f \leq M$$ $$f' \geq C \geq K_1 + K_2M$$ where $K_1$ and $K_2$ are fixed positive constants and are given. Is it possible ...
2
votes
1answer
32 views

For what functions does this inequality hold?

I would like to know for what kind of functions, the inequality $$\frac{f^\prime(x)}{f(x)}\leq \frac{1}{b-a}$$ hold for all $x\in[a,b]$. $f^\prime(x)$ is derivative w.r.t. $x$. Is there some general ...
4
votes
1answer
78 views

Increasing marginal product implies increasing returns to scale?

Setup Let $f(x,y)$ be twice differentiable in both $x$ and $y$. Assume $\partial f/\partial x>0,\partial f/\partial y>0$ for $x,y>0$. $f$ is said to have increasing marginal product of ...
1
vote
2answers
83 views

Proving $f(x)\leq x$ with some conditions

Let $f:[0,1]\to [0,1]$ be a function such that $f(1)=1$ $f(x)+f(y)\leq f(x+y)$, for any numbers $x$ , $y$ , $x+y \in [0,1]$ Then we have to show that $f(x)\leq x$ for any $x\in [0,1]$. I can ...
0
votes
1answer
17 views

Increasing rate of a continuous function

Consider $f: X \rightarrow X$ continuous, with $X \subset \mathbb{R}^n$ compact convex. I am wondering on conditions on $f$ so that there exists $\epsilon > 0$ such that $$ (x-y)^\top \left( f(x) ...
0
votes
0answers
18 views

Get a count of a set when a condition is met

This feels like an easy question, but I can't seem to hammer it down. Just like $\Sigma$ can be used to sum over several items, it there a way to count them? The simple example, is that I want to ...
2
votes
1answer
39 views

I need help showing this inequality

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a twice differentiable function such that $f'>0$, $f''<0$, and $f(0)=0$. I need to show, that for every $x>0$: $\frac{f(x)}{f'(x)}>x$ Thanks ...
0
votes
1answer
40 views

Inferring a characteristic of a ratio of functions from the ratio of their derivatives

This is a strange one, but I need help trying to understand whether there is any logic behind this or not. Given $\frac {f(\sqrt{2})}{g(\sqrt{2})}=2$, and $\frac {f'(x)}{g'(x)}>2$ for all ...
1
vote
2answers
57 views

Is this function inequality true?

Let $\lambda$ and $\lambda_L$ be the values of the function $f(x,y)$ at the optimum for problems \begin{align} \lambda=\max_{x}\min_{y}f(x,y) \end{align} \begin{align} ...
0
votes
1answer
17 views

Inequality Conditions

Let $h_{k}(x)>0$ and $\sum_{k=1}^{l}h_{k}(x)=1$ (Here, $h_{k}(x)$ are some continuous functions). Is the statement below correct or not? $f_{k}(x)<0$ when $g_{k}(x)=0$, $\forall x \neq 0$, ...
4
votes
5answers
277 views

Prove that $\forall x>0, \frac {x-1}{\ln(x)} \geq \sqrt{x} $.

This inequality arose in this question Prove that : $|f(b)-f(a)|\geqslant (b-a) \sqrt{f'(a) f'(b)}$ with $(a,b) \in \mathbb{R}^{2}$ : $$\forall x>0, \frac {x-1}{\ln(x)} \geq \sqrt{x} $$ ...
11
votes
2answers
269 views

$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
1
vote
2answers
41 views

Prove $(x+y)^a\leq x^a+y^a$ if $0<a\leq1$ and $x,y\geq0$

Prove $(x+y)^a\leq x^a+y^a$ if $0<a\leq1$ and $x,y\geq0$ I need to prove this step for a bigger question. It should be quite basic but I just have no idea...
5
votes
2answers
127 views

Show that $f(x)=0$ for all $x\geq0$

I have been struggling with this problem.. Q. Let $f(x)$, $x\geq 0$, be a non-negative continuous function, and let $F(x)=\int_0^x f(t) dt$, $x\geq0$. If for some $c>0$, $f(x)\leq cF(x)$ for ...
0
votes
1answer
37 views

Functional inequality : bounded functions

Suppose $f''(x)$ exists ($f(x)$ can be differentiated two times) And the function and the second derivative is bounded : $\left|f(x)\right|\le P$, $\left|f''(x)\right|\le Q$ Then, how can I prove ...
1
vote
2answers
90 views

prove: $x^\alpha - \alpha x \le 1-\alpha$

$0 < \alpha < 1,\forall x \ge 0$ prove: $x^\alpha - \alpha x \le 1-\alpha$ What I did: We know that: $${x^\alpha } = {e^{\ln ({x^\alpha })}} = {e^{\alpha \ln (x)}}$$ Therefore, we need to ...
2
votes
5answers
58 views

More precise way of solving inequality

I need to solve this function: $$ \lvert x^2-1\rvert\ge 2x-2\\ $$ I solved this equation: For $x<0$, the solution is non existing, here I got negative root, when I tried to solve quadratic ...
5
votes
1answer
62 views

function inequality $f(x+y)+y \leq f(f(f(x)))$

$f(x+y)+y \leq f(f(f(x)))$ find all possible solution for $ f: \mathbb {R} \rightarrow \mathbb {R}$
19
votes
1answer
381 views

Existence of two real numbers satisfying $f(x-f(y))>yf(x)+x$

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be a function. Is it always the case that for some $x,y \in \mathbb R$, the inequality $f(x-f(y))>yf(x)+x$ holds? Thanks in advance.
2
votes
1answer
19 views

Find a minium value of a function

Given x,y,z are positive real numbers such that $$ x^2+y^2+6z^2=4z(x+y). $$ Find the minimum value of the following function $$ P=\frac{x^3}{y(x+z)^2}+\frac{y^3}{x(y+z)^2}+\frac{\sqrt{x^2+y^2}}{z} $$
0
votes
2answers
91 views

Is $L_\infty$ norm the smallest or largest?

I am a little bit confused. For a $L_p$ function norm, is it true that for any $ p<\infty $, $$ \|f\|_p>\|f\|_\infty$$ Is the statement true for any domain? I want to know more inequality about ...
0
votes
2answers
34 views

Prove the lines given by the functions

There is the second problem of the day that I have been stuck on for quite some time, and I am having trouble examining how to evaluate this equation to simple form. Prove the lines given by the ...
1
vote
3answers
44 views

Prove directly that for $x,y \ge 0$

The question that I am stuck on is as follows: Prove directly that for $x,y \ge 0$ $\sqrt{xy}\le (x+y)/2$. When does equality hold? I have been working at it for almost 20 minutes. Can ...
2
votes
1answer
124 views

Proving the inequality $\frac{\tan{x}}{x} > \frac{x}{\sin{x}}$ [duplicate]

I'm trying to prove this inequality: $$\frac{\tan{x}}{x} > \frac{x}{\sin{x}} $$ for all $x$ in $(0,\frac{\pi}{2})$. I tried analyzing the derivates, but that's just making it more complicated. Any ...
1
vote
1answer
115 views

Proving that linear combination of exponentials is positive

I found the following question in a book without any proof. Question : Prove that $$f(t)=3-5e^{-2t}+6e^{-3t}+2e^{-5t}-3e^{-(3-\sqrt5)t}-3e^{-(3+\sqrt5)t}\gt0$$ for any $t\gt0$. The book says that ...
1
vote
4answers
92 views

The sine inequality $\frac2\pi x \le \sin x \le x$ for $0<x<\frac\pi2$

There is an exercise on $\sin x$. How could I see that for any $0<x< \frac \pi 2$, $\frac 2 \pi x \le \sin x\le x$? Thanks for your help.
1
vote
2answers
729 views

Solving the domain and range of a region satisfying two inequalities?

The question I was provided was: "Find the domain and range of the region satisfied by the following inequalities: i) $y \ge (x-1)^2$ ii)$y \le2x+1$ Any help would be greatly appreciated. Would you ...
2
votes
1answer
33 views

Condition for differential inequality

Let $f(x) = \frac{e^{ - ax}}{1 + {e^{bx}}}$, where $x>0$, $a$ and $b$ are positive constants. Find the condition of $a$ and $b$ so that $$ ( - 1)^nf^{(n)}(x) \ge 0 $$ with all $x>0$ and $n$, ...
2
votes
1answer
63 views

Need help showing the supremum of a function exists.

I was wondering if anyone knows a technique for proving that this function has a supremum less than infinity for $x \in \mathbb{R}$ ,$x \in [-1,1]$ (I am very certain that it does). The function is, ...
5
votes
2answers
157 views

Prove that $-\frac{\sqrt{x}}{1+x}\log{x} \leq \log{2}$ for $0 < x < 1$

Graphically and numerically it is obvious but I'm looking for an analytical reasoning. Just maximizing the left hand side does not yield an analytical expression for the maximum. I also tried some ...
0
votes
3answers
175 views

Where is $f(x) = \log(5x^2-8x-4)+\sqrt{x-1}$ defined?

Find the values of $x$ for which function is defined: $f(x) = \log(5x^2-8x-4)+\sqrt{x-1}$. $ \log(5x^2-8x-4) > 0 \Rightarrow 5x^2-8x-4 > 1 \Rightarrow 5x^2-8x-5 > 0 $ $x = \frac{ 8 \pm ...
1
vote
1answer
40 views

If there is a $T$ such that $V(t)<V(t-T) \ \forall t$, does that imply $V(t) \to 0$?

Let $V(t)$ denote a continuous scalar function $\mathbb{R} \mapsto \mathbb{R}$. Assume that we can find a constant $T \in \mathbb{R}$ such that $V(t)<V(t-T)$ for all $t$. Does that imply that $V(t) ...
5
votes
1answer
68 views

How to show $f(x) \leq 1+\frac{\pi}{4}$ for every $x \geq 1$

Suppose $f$ is a real-valued differentiable function defined on $[ 1,\infty)$ with $f(1)=1$. Suppose , moreover , that $f$ satisfies $$f'(x)=\frac{1}{x^2+f^2(x)}$$ Show that $f(x) \leq ...
2
votes
2answers
78 views

Inequalities of Integer functions

I have the following statement that I'm trying to prove: Assume that $f,g: \mathbb{N} \rightarrow \mathbb{R}^{\ge0}$. If $f(n) \ge g(n)$ then $\lceil f(n) \rceil \ge \lceil g(n) \rceil $. I have a ...
-2
votes
1answer
289 views

Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $ [closed]

Let $f\left( x\right)$ be a $C^{2}$ function on $\mathbb{R}$. Show that $$\sup \left| f'\left( x\right) \right| ^{2}\leqslant4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) ...
0
votes
0answers
39 views

Extracting a function from set of inequalities

I have set of inequalities in two dimension space which represent relation between $X$ and $Y$. now I want a function whose input is $X$ and output is $Y$. In other words, I want $F$ such that ...
-1
votes
1answer
57 views

Exercise of functions of a real variable

Let $f, g\colon\mathbb{R}\rightarrow\mathbb{R}$ functions so that for all $\,x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)g(y).$$ Prove that if $f$ is not constant zero function and $|f(x)|\leqslant ...
2
votes
1answer
77 views

Simple inequality help

I need a function $f(x)$ that satisfies the properties bellow for all integers $k$ $$ \frac{\log(k+1)}{k+1}-\log\left(1+\frac 1 k\right)+f(k+1)-f(k)<0 \ $$ $$ \lim_{k \rightarrow \infty} f(k)=0 $$ ...
3
votes
2answers
204 views

Ceiling function inequality

In class, we used the fact that $\lceil{a + b \rceil} \geq \lceil{a}\rceil + \lfloor{b}\rfloor$. However, we weren't given a proof of this statement. I am interested to see how this works. Can anyone ...
1
vote
2answers
61 views

Smooth function on $\mathbb R$ whose small increments are not controlled by the first derivative at infinity

I need some help in finding a (as simple as possible) smooth function $f:\mathbb R \rightarrow \mathbb R$ which does NOT satisfy the following: There exist a constant $C>0$, a compact ...
1
vote
1answer
44 views

how to find arrange the following functions in increasing or decreasing order?

I have the following three functions $f_1(x) = \frac{1}{4} (8-3x + \sqrt{(x-2) (5x-14)}) (1-x)$ $f_2(x) = \frac{1}{8} (12-4x + \sqrt{2} \sqrt{(5x-14)(x-3)} + \sqrt{2} \sqrt{(x-2)(x-3)} )(1-x)$ ...
2
votes
1answer
132 views

Upper bound for $\Gamma(x+y)$

Let $x, y \geq 1$ be two real numbers. I am wondering if one can get an upper bound for $\Gamma(x+y)$ in terms of $\Gamma(x)\Gamma(y)$? Any references or ideas are very appreciated. Thank you.
3
votes
1answer
190 views

Minimizing a function over two variables

Given two natural numbers $i$ and $p$ such that $0 < i \leqslant 2^p$, let $$ \psi(p,i) := p - \alpha + 1 - \frac{1}{2^p}\left((2^p+i)\lg(2^p+i) - i\lg i - i + \alpha - \frac{2^p}{i+1} - ...
0
votes
1answer
47 views

Mathematical function for four corelated attributes

I have $4$ attributes $A,B,C,D$ each of them takes value between $[0,1]$ The more $A$ and $B$, the more the function value is. The more $C$ and $D$, the less the function value is. if C or D equals ...
0
votes
2answers
64 views

A function that maps inequalities to $-1$, $0$, or $1$.

The Python computer language has a built-in operation cmp(a,b) that returns $-1$, $0$ or $1$, if $a<b$, $a=b$ or $a>b$, respectively. I'd like to know if ...
2
votes
2answers
73 views

Geometric intuition for the inequality $(f(y) - c) ( y - d ) \geq (f(d) - c) ( f^{-1}(c) - d )$

Good day to everyone. I am interested in the geometric intuition for the following statement: Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a monotonically increasing, invertible function and $c,d \in ...