what do you want to say with "find the range of b"? can you explain, please?

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?

min(b,2sqrt(|a(a-b)|))=3 take cases and solve

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!

Obviosly $b=f(0)\ge 3$. If $b=3$, then $a\le \frac 32 $(any). If $b>3$, then $a=\frac{b-\sqrt{b^2-9}}{2}.$

Here is my solution.

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