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2

$begingroup$

Let $f(x)$ be defined and continuous and derivable for $x>-1$,
$f(0)=1$, $f’(0)=0$ and
$$f''(x) = frac {1+x}{1+f(x)}.$$
Prove that $f’(x)$ is concave up for all $x>-1$.

My attempt:
I tried to integrate by multiplying both sides by $dy/dx$
but could not proceed further.

ordinary-differential-equations derivatives monotone-functions

share|cite|improve this question

edited 1 hour ago

Robert Z

99.7k1068140

asked 1 hour ago

mavericmaveric

85712

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is my editing correct?
  $endgroup$
  – Robert Z
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is "concave up"? And the assumption sholud be that $f$ is twice differentiable.
  $endgroup$
  – Paul Frost
  1 hour ago
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @PaulFrost: Some people use the terms “concave up/down” for “convex/concave.”
  $endgroup$
  – Martin R
  1 hour ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Okay, but in my opinion it is unusual notation. You should at least explain this in your question and especially in the title.
  $endgroup$
  – Paul Frost
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It is very likely that in this question you're not expected to find $f$ or $f'$ explicitly. Can you start proving that $f$ is positive, on what interval? Then you can consider proving that $f''$ is monotone.
  $endgroup$
  – Joce
  45 mins ago
  
  
  

  
  
  
  

 | 
show 1 more comment

2

$begingroup$

Let $f(x)$ be defined and continuous and derivable for $x>-1$,
$f(0)=1$, $f’(0)=0$ and
$$f''(x) = frac {1+x}{1+f(x)}.$$
Prove that $f’(x)$ is concave up for all $x>-1$.

My attempt:
I tried to integrate by multiplying both sides by $dy/dx$
but could not proceed further.

ordinary-differential-equations derivatives monotone-functions

share|cite|improve this question

edited 1 hour ago

Robert Z

99.7k1068140

asked 1 hour ago

mavericmaveric

85712

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is my editing correct?
  $endgroup$
  – Robert Z
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is "concave up"? And the assumption sholud be that $f$ is twice differentiable.
  $endgroup$
  – Paul Frost
  1 hour ago
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @PaulFrost: Some people use the terms “concave up/down” for “convex/concave.”
  $endgroup$
  – Martin R
  1 hour ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Okay, but in my opinion it is unusual notation. You should at least explain this in your question and especially in the title.
  $endgroup$
  – Paul Frost
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It is very likely that in this question you're not expected to find $f$ or $f'$ explicitly. Can you start proving that $f$ is positive, on what interval? Then you can consider proving that $f''$ is monotone.
  $endgroup$
  – Joce
  45 mins ago
  
  
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Let $f(x)$ be defined and continuous and derivable for $x>-1$,
$f(0)=1$, $f’(0)=0$ and
$$f''(x) = frac {1+x}{1+f(x)}.$$
Prove that $f’(x)$ is concave up for all $x>-1$.

My attempt:
I tried to integrate by multiplying both sides by $dy/dx$
but could not proceed further.

ordinary-differential-equations derivatives monotone-functions

share|cite|improve this question

edited 1 hour ago

Robert Z

99.7k1068140

asked 1 hour ago

mavericmaveric

85712

$endgroup$

share|cite|improve this question

edited 1 hour ago

Robert Z

99.7k1068140

asked 1 hour ago

mavericmaveric

85712

share|cite|improve this question

edited 1 hour ago

Robert Z

99.7k1068140

asked 1 hour ago

mavericmaveric

85712

edited 1 hour ago

Robert Z

99.7k1068140

edited 1 hour ago

Robert Z

99.7k1068140

asked 1 hour ago

mavericmaveric

85712

asked 1 hour ago

mavericmaveric

85712