If $\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3$, then (1) Find the range of $b$; (2) for every given $b$, find $a$.
Problem
Source: China South East Mathematical Olympiad2011
Tags: function, calculus, derivative, algebra proposed, algebra
18.08.2011 14:33
what do you want to say with "find the range of b"? can you explain, please?
18.08.2011 15:13
i think he means that the expression holds for all real numbers .the minima is 3 .then what possible values can b have?isn't it?
19.08.2011 10:32
min(b,2sqrt(|a(a-b)|))=3 take cases and solve
19.08.2011 10:43
lssl wrote: If $\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3$, then (1) Find the range of $b$; (2) for every given $b$, find $a$. lssl,just by a simple derivative,one can solve it!
22.08.2011 13:00
Obviosly $b=f(0)\ge 3$. If $b=3$, then $a\le \frac 32 $(any). If $b>3$, then $a=\frac{b-\sqrt{b^2-9}}{2}.$