The simplest function that fits seems to be the quadratic f(x)=x^22x+2.
But then we require (c^22c+2)(2c2)+2=0 which is a cubic with solutions around {.9, 1.3, 2.4}. There are none on the interval (0,1). This implies the there are atleast zero c.
When I bumped up to a cubic there is more wiggle room. f(x)=ax^3+(1a)x^22x+2 fits the qualifications for any a.
Playing with Desmos, I can find values of a with 2 solutions (if a=0.5 then c=0.206 or c=0.791) or 1 solution (if a=0.5 then c=0.347) but I cannot find where there are zero values of c.
A question on notation then: Does f:ℝ→ℝ rule out quadratic equations?

Posted by Jer
on 20190625 09:17:38 