Find all open intervals on which the function $\displaystyle f(x) = \frac {x^2} {x^2 + 4}$ is decreasing
Hello, Samantha!
Exactly where is your difficulty?
. . You don't know the Quotient Rule?
. . You don't understand increasing/decreasing?
. . You can't handle inqualities?
Quotient Rule: .$\displaystyle f'(x) \;=\;\frac{(x^2+4)\cdot2x - x^2\cdot2x}{(x^2+4)^2}\;=\;\frac{8x}{(x^2+4)^2}$Find all open intervals on which the function $\displaystyle f(x) \:=\: \frac {x^2} {x^2 + 4}$ is decreasing
The function is decreasing where $\displaystyle f'(x) \,<\,0$
. . So we have: .$\displaystyle \frac{8x}{(x^2+4)^2} \:<\:0$
Since the denominator is always positive, we have: .$\displaystyle 8x \:<\:0\quad\Rightarrow\quad x \:<\:0$
Therefore, $\displaystyle f(x)$ is decreasing on the interval: .$\displaystyle (-\infty,\:0)$