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A family of functions $\{f_{a,b} \mid a, b \in \mathbb{R}\}$ is given thru $$f_{a,b}(x)=\frac{x^4+ax^2+b}{x^2+1}.$$ It is asked to pick those functions such that their graph $\Gamma_{a,b}$ is tangent to the horizontal axis in two distinct points.

I have written the conditions of tangency and of passage thru a point $(x,0)$. The discussion is a bit long, but possible. I wonder if there is any "shortcut", i.e. some approach which leads to the solution more quickly.

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    What tangency condition have you established?2012-04-29
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    The derivative of the function must vanish at two distinct points.2012-04-29
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    Put $x^2:=u$ and construct tangencies $(u,0)$ with $u>0$.2012-04-29
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    You should have that $f$ and $f'$ are simultaneously zero. This should give you two polynomial equations, and the condition you need is that they have a common factor - if all else fails, you could use the division algorithm for polynomials. Note that $f$ is an even function - evidently if $x$ is a solution so is $-x$, so most of the time you only need to find one point - another comes automatically. Christian's hint uses this to simplify the calculations. There is a special case $x=0$ to consider.2012-04-29

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