Prove that a function is monotically increasing.

Question

can someone give me a hint on how to prove f(x)=(3x+1)/(4-x^2) , for all x, except x=-2 and x=2 that is monotically increasing? i know the definitions but i cannot prove it for some reason.

Answer 1

Hint: given $f(x)=\dfrac{3x+1}{4-x^2}$, we have $$f'(x)=\frac{3(4-x^2)+2x(3x+1)}{(4-x^2)^2}=\frac{3x^2+2x+12}{(4-x^2)^2}.$$

How can we show that we have $f'(x)>0\ \forall x\in\Bbb R, x\neq\pm2$?

Answer 2

HINT: $$\frac{3x^2+2x+12}{(4-x^2)^2}>0\underbrace{\Longleftrightarrow}_{x\neq\pm2}\frac{35}{3}+3(\frac{1}{3}+x)^2>0$$