# Tagged Questions

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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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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) ... 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 ... 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 ... 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 ... 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} ... 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, ... 5answers 272 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} $$... 2answers 266 views ### \lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor Prove that: \lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor, for n > 1,\, n\in \mathbb{N} For example. For n=2, we have \lfloor ... 2answers 41 views ### Prove (x+y)^a\leq x^a+y^a if 0<a\leq1 and x,y\geq0 Prove (x+y)^a\leq x^a+y^a if 0<a\leq1 and x,y\geq0 I need to prove this step for a bigger question. It should be quite basic but I just have no idea... 2answers 127 views ### Show that f(x)=0 for all x\geq0 I have been struggling with this problem.. Q. Let f(x), x\geq 0, be a non-negative continuous function, and let F(x)=\int_0^x f(t) dt, x\geq0. If for some c>0, f(x)\leq cF(x) for ... 1answer 37 views ### Functional inequality : bounded functions Suppose f''(x) exists (f(x) can be differentiated two times) And the function and the second derivative is bounded : \left|f(x)\right|\le P, \left|f''(x)\right|\le Q Then, how can I prove ... 2answers 89 views ### prove: x^\alpha - \alpha x \le 1-\alpha 0 < \alpha < 1,\forall x \ge 0 prove: x^\alpha - \alpha x \le 1-\alpha What I did: We know that:$${x^\alpha } = {e^{\ln ({x^\alpha })}} = {e^{\alpha \ln (x)}}$$Therefore, we need to ... 5answers 58 views ### More precise way of solving inequality I need to solve this function:$$ \lvert x^2-1\rvert\ge 2x-2\\ $$I solved this equation: For x<0, the solution is non existing, here I got negative root, when I tried to solve quadratic ... 1answer 62 views ### function inequality f(x+y)+y \leq f(f(f(x))) f(x+y)+y \leq f(f(f(x))) find all possible solution for  f: \mathbb {R} \rightarrow \mathbb {R} 1answer 379 views ### Existence of two real numbers satisfying f(x-f(y))>yf(x)+x Let f:\mathbb{R} \longrightarrow \mathbb{R} be a function. Is it always the case that for some x,y \in \mathbb R, the inequality f(x-f(y))>yf(x)+x holds? Thanks in advance. 1answer 19 views ### Find a minium value of a function Given x,y,z are positive real numbers such that$$ 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} $$2answers 91 views ### Is L_\infty norm the smallest or largest? I am a little bit confused. For a L_p function norm, is it true that for any  p<\infty ,$$ \|f\|_p>\|f\|_\infty$$Is the statement true for any domain? I want to know more inequality about ... 2answers 34 views ### Prove the lines given by the functions There is the second problem of the day that I have been stuck on for quite some time, and I am having trouble examining how to evaluate this equation to simple form. Prove the lines given by the ... 3answers 44 views ### Prove directly that for x,y \ge 0 The question that I am stuck on is as follows: Prove directly that for x,y \ge 0 \sqrt{xy}\le (x+y)/2. When does equality hold? I have been working at it for almost 20 minutes. Can ... 1answer 124 views ### Proving the inequality \frac{\tan{x}}{x} > \frac{x}{\sin{x}} [duplicate] I'm trying to prove this inequality:$$\frac{\tan{x}}{x} > \frac{x}{\sin{x}} $$for all x in (0,\frac{\pi}{2}). I tried analyzing the derivates, but that's just making it more complicated. Any ... 1answer 109 views ### Proving that linear combination of exponentials is positive I found the following question in a book without any proof. Question : Prove that$$f(t)=3-5e^{-2t}+6e^{-3t}+2e^{-5t}-3e^{-(3-\sqrt5)t}-3e^{-(3+\sqrt5)t}\gt0$$for any t\gt0. The book says that ... 4answers 91 views ### The sine inequality \frac2\pi x \le \sin x \le x for 0<x<\frac\pi2 There is an exercise on \sin x. How could I see that for any 0<x< \frac \pi 2, \frac 2 \pi x \le \sin x\le x? Thanks for your help. 2answers 670 views ### Solving the domain and range of a region satisfying two inequalities? The question I was provided was: "Find the domain and range of the region satisfied by the following inequalities: i) y \ge (x-1)^2 ii)y \le2x+1 Any help would be greatly appreciated. Would you ... 1answer 33 views ### Condition for differential inequality Let f(x) = \frac{e^{ - ax}}{1 + {e^{bx}}}, where x>0, a and b are positive constants. Find the condition of a and b so that$$ ( - 1)^nf^{(n)}(x) \ge 0 $$with all x>0 and n, ... 1answer 62 views ### Need help showing the supremum of a function exists. I was wondering if anyone knows a technique for proving that this function has a supremum less than infinity for x \in \mathbb{R} ,x \in [-1,1] (I am very certain that it does). The function is, ... 2answers 156 views ### Prove that -\frac{\sqrt{x}}{1+x}\log{x} \leq \log{2} for 0 < x < 1 Graphically and numerically it is obvious but I'm looking for an analytical reasoning. Just maximizing the left hand side does not yield an analytical expression for the maximum. I also tried some ... 3answers 170 views ### Where is f(x) = \log(5x^2-8x-4)+\sqrt{x-1} defined? Find the values of x for which function is defined: f(x) = \log(5x^2-8x-4)+\sqrt{x-1}.  \log(5x^2-8x-4) > 0 \Rightarrow 5x^2-8x-4 > 1 \Rightarrow 5x^2-8x-5 > 0  x = \frac{ 8 \pm ... 1answer 40 views ### If there is a T such that V(t)<V(t-T) \ \forall t, does that imply V(t) \to 0? Let V(t) denote a continuous scalar function \mathbb{R} \mapsto \mathbb{R}. Assume that we can find a constant T \in \mathbb{R} such that V(t)<V(t-T) for all t. Does that imply that V(t) ... 1answer 68 views ### How to show f(x) \leq 1+\frac{\pi}{4} for every x \geq 1 Suppose f is a real-valued differentiable function defined on [ 1,\infty) with f(1)=1. Suppose , moreover , that f satisfies$$f'(x)=\frac{1}{x^2+f^2(x)}$$Show that f(x) \leq ... 2answers 78 views ### Inequalities of Integer functions I have the following statement that I'm trying to prove: Assume that f,g: \mathbb{N} \rightarrow \mathbb{R}^{\ge0}. If f(n) \ge g(n) then \lceil f(n) \rceil \ge \lceil g(n) \rceil . I have a ... 1answer 288 views ### Prove \sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right|  [closed] Let f\left( x\right) be a C^{2} function on \mathbb{R}. Show that$$\sup \left| f'\left( x\right) \right| ^{2}\leqslant4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) ...
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 ...