MHB Increase/Decrease of Rational Function | Jillian's Yahoo Answers

[MarkFL]

Here is the question:

How to find the intervals of increase and decrease?

F(x)= x^2 - 1/ x^2 + 1

Here is a link to the question:

How to find the intervals of increase and decrease? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[MarkFL]

Hello Jillian,

I am assuming we have:

$$f(x)=\frac{x^2-1}{x^2+1}$$

To investigate where the function is increasing/decreasing, we need to compute the first derivative, and find where it is positive (function increasing) and where it is negative (function decreasing).

Using the quotient and power rules of differentiation, we find:

$$f'(x)=\frac{(x^2+1)(2x)-(x^2-1)(2x)}{(x^2+1)^2}=\frac{2x(x^2+1-x^2+1)}{(x^2+1)^2}=\frac{4x}{(x^2+1)^2}$$

Now, we see the denominator is positive for all real $x$, so we need only concern ourselves with the sign of the numerator, and we see this simply has the sign of $x$ itself. Hence:

$$(-\infty,0)$$ $f(x)$ is decreasing.

$$(0,\infty)$$ $f(x)$ is increasing.

To Jillian and any other guests viewing this topic, I invite and encourage you to post other calculus questions here in our http://www.mathhelpboards.com/f10/ forum.

Best Regards,

Mark.