All Questions
5
questions
1
vote
2
answers
72
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Prove that $0 < g(x) - \ln x \leq \ln\left(\frac{x+2}{x}\right)$ for $x > 0$
$$g(x) = \ln(x + 1 + e^{-x})$$
My question is prove that $0 < g(x) - \ln x \leq \ln\left(\frac{x+2}{x}\right)$ for $x > 0$
How do I do that?
My attempts:
I have only successfully proved the ...
- 433
0
votes
1
answer
47
views
Prove that $\left | f(x) - a \right | \leq \frac{1}{2} \left | x - a \right |$
$f(x) = 1 - \frac{1}{x}(\sqrt{1 + x^2}-1)$
$|f'(x)| \leq \frac{1}{2}$
$a$ is a solution for $f(x) = x$ where $0.65 < a < 0,7$
The question says:
Prove that $\left | f(x) - a \right | \leq \frac{...
- 433
-2
votes
2
answers
116
views
Is it true that $\sqrt{ab}\le \frac{a-b}{\ln a - \ln b}$ for any $a\neq b>0$? [closed]
Is it true that $\sqrt{ab}\le \frac{a-b}{\ln a - \ln b}$ for any $a\neq b>0$?
If so, any thoughts on how to prove this?
- 431
2
votes
0
answers
661
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Monotonically increasing function and Lipschitz continuous functions
Let $f(t,x):[0,T]\times \mathbb{R}\rightarrow \mathbb{R}$.
If $$|f(t,x)-f(t,y)|\leq C|x-y|, C>0$$ (Lipschitz continuous functions).
I need to found a monotonically increasing function $g(t;x)$ ...
- 73
0
votes
3
answers
192
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Relationship $f(x)$ and $\max(f(x))$
If $f(x),g(x)$ are functions from $\Bbb R$ to $\Bbb R$ and we define $$X=\max\left[0,\max(-f(x))\right]$$ and $$Y=\max\left[0,\max(-g(x))\right]$$
I need to know if the following inequality is true: ...
- 73