text: # Lower bound over a concave function
  
  Hi everyone, I will be to grateful if help me find a tight lower bound $g(x)$ over the following concave function: $$f(x) = \sqrt{1+4x} -1 + \log(\sqrt{1+4x}-1) - \log(2x) \geq g(x),$$ where $x \geq 0$. The taylor expansion around the point ($x = 0$) of this function is given by: $$f(x) = x - \frac{x^2}{2} - + \frac{2x^3}{3} - \frac{5x^4}{4} + \frac{14x^{5}}{5} - \ldots$$
  
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  What do require exactly on the function $g$? –  Pietro Majer May 30 '11 at 17:59
  To be quite tight, and preferably $g(x)=ax+b$ where $b \geq 1$. –  Farzad May 30 '11 at 18:37
  Is $g(x)$ of interest near 0, for large values, or somewhere in between? For large $x$, the function is dominated by the $\sqrt{4x+1}$ term so a linear bound is going to be poor; you need to define the range in which the bound is of interest. If the linearity requirement is dropped, clearly $g(x) = 2\sqrt{x} - 1$ is going to be a fairly good lower bound for all large enough $x$. –  András Salamon Sep 23 '13 at 15:19
  and $ax+b$ with $b\geq1$ is impossible near zero, from your taylor expansion. –  Jung Wen Chen Jan 29 '14 at 14:49
explanation: The text extract is a discussion about finding a tight lower bound for a given concave function. It includes a mathematical function and its Taylor expansion. The extract also contains a question asking for help in finding a tight lower bound \( g(x) \) for the function \( f(x) \). Additionally, there are comments from other users providing feedback and asking for clarification on the requirements for \( g(x) \).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Lower bound over a concave function
  
  Hi everyone, I will be to grateful if help me find a tight lower bound $g(x)$ over the following concave function: $$f(x) = \sqrt{1+4x} -1 + \log(\sqrt{1+4x}-1) - \log(2x) \geq g(x),$$ where $x \geq 0$. The taylor expansion around the point ($x = 0$) of this function is given by: $$f(x) = x - \frac{x^2}{2} - + \frac{2x^3}{3} - \frac{5x^4}{4} + \frac{14x^{5}}{5} - \ldots$$
  
  -
  What do require exactly on the function $g$? –  Pietro Majer May 30 '11 at 17:59
  To be quite tight, and preferably $g(x)=ax+b$ where $b \geq 1$. –  Farzad May 30 '11 at 18:37
  Is $g(x)$ of interest near 0, for large values, or somewhere in between? For large $x$, the function is dominated by the $\sqrt{4x+1}$ term so a linear bound is going to be poor; you need to define the range in which the bound is of interest. If the linearity requirement is dropped, clearly $g(x) = 2\sqrt{x} - 1$ is going to be a fairly good lower bound for all large enough $x$. –  András Salamon Sep 23 '13 at 15:19
  and $ax+b$ with $b\geq1$ is impossible near zero, from your taylor expansion. –  Jung Wen Chen Jan 29 '14 at 14:49
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a discussion about finding a tight lower bound for a given concave function. It includes a mathematical function and its Taylor expansion. The extract also contains a question asking for help in finding a tight lower bound \( g(x) \) for the function \( f(x) \). Additionally, there are comments from other users providing feedback and asking for clarification on the requirements for \( g(x) \).
  
  Label: question-and-answer