5. Try again Get help Practice similar calculated from the equation \[ \frac{1}{R}=\frac{1}{r}+\frac{1}{s}, \] where \( R, r \), and \( s \) are positive. Assume that \( s \) is constant. Find \( \frac{d R}{d r} \) : \( \frac{d R}{d r}=\frac{s^{2}}{(r+s)^{2}} \) Is \( R \) and increasing or decreasing function of \( r \) ? incre asin \( g \) (Enter increasing, decreasing, neither, or both (write both if there are values of \( r \) for which \( R \) is increasing, and other values for which it is decreasing; enter neither if this is a constant function.) If we consider the interval \( a \leq r \leq b \), where does \( R \) take on its global maximum and minimum values? maximum: \( r=a \) minimum: \( r=b \) (Enter none if there is no global maximum or minimum for this

Let \( f(x)=\frac{x}{x^{2}+13} \). Find (a) the intervals on which \( f \) is increasing, (b) the intervals on which \( f \) is decreasing, (c) the open intervals on which \( f \) is concave up, (d) the open intervals on which \( f \) is concave down, and (e) the \( x \)-coordinates of all inflection points. (a) \( f \) is increasing on the interval(s) \( (-\sqrt{13}, \sqrt{13}) \) (b) \( f \) is decreasing on the interval(s) \( (-\mathrm{inf},-\sqrt{13}) U(\sqrt{13} \), inf) (c) \( f \) is concave up on the open interval(s) \( (0 \), inf) (d) \( f \) is concave down on the open interval(s) (-inf, 0\( ) \) (e) the \( x \) coordinate(s) of the points of inflection are 0 Notes: In the first four boxes, your answer should either be a single interval, such as [0,1), a comma separated list of intervals, such as (-inf, 2), (3,4], or the word "none". In the last box, your answer should be a comma separated list of \( x \) values or the word "none".

derivates \( y=\frac{1+x^{3}}{1-x^{3}} \)

\( y=e^{2 \cos 2 x} \) deviuates

(f) \( \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\cos x}{1-\tan x} \)

(f) \( \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\cos x}{1-\tan x} \) (g) \( \lim _{x \rightarrow 0} \frac{x-\sin 2 x}{x+\sin 3 x} \) (h) \( \lim _{x \rightarrow 0} \frac{1-\sqrt{\cos x}}{x^{2}} \) (i) \( \lim _{x \rightarrow \infty}\left(\frac{x-1}{x+3}\right)^{x+2} \)