Find the highest and lowest value of the next function without the use of derivatives
$$f(x)={3x^2+6x+6\over x^2+4x+5}$$
Okay so my teacher gave me this problem, and told me to strictly solve it without using derivatives. Obviously with using derivatives it's pretty easy, just find where $f'(x_0)=0$, and check it's second derivative in $x_0$ to see if it's a maximum or minimum.
So I was thinking about this:
Since this function is written as a fraction, the lowest possible value should be when the numerator value is the lowest, while the denominator is the highest possible value.
BUT, again to find the highest and lowest value of both I could use derivatives again, which again isn't the point.
I can see that both the numerator and denominator of this fraction is always positive for every $x$, so maybe there's a hint that I can't see there?
Maybe I could use limits too, but I'm not sure how.
Best Answer
If $\kappa$ belongs to the range of $f$, it follows that $$3x^2+6x+6 = \kappa(x^2+4x+5)$$ has a real solution, hence the discriminant of $$ (3-\kappa)x^2+(6-4\kappa)x+(6-5\kappa) $$ i.e. $-4\kappa^2+36\kappa-36$, is non-negative. It follows that the extreme values of $f$ are given by the roots of $-4\kappa^2+36\kappa-36$, i.e. $\color{red}{\frac{3}{2}\left(3\pm\sqrt{5}\right)}$.
I believe this (exploiting discriminants) was the ancient method for the determination of the stationary values of rational functions, before the Method of Fluxions. But relics have their elegance!