When an electric current passes through two resistors with resistance \( r \) and \( s \), connected in parallel, the combined resistance, \( R \), can be calculated from the equation where \( R, r \), and \( s \) are positive. Assume that \( s \) is constant. Find \( \frac{d R}{d r} \) : \( \frac{d R}{d r}= \) Is \( R \) and increasing or decreasing function of \( r \) ? (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: \( r= \) minimum: \( r= \) (Enter none if there is no global maximum or minimum for this

\( \left. \begin{array} { l } { f ^ { \prime } ( x ) = } \\ { f ( x ) = \sqrt { \cos ( e ^ { x ^ { 5 } \sin ( x ) } ) } } \\ { \frac { 5 x ^ { 4 } e ^ { x ^ { 5 } \cdot \sin ( x ) } \cdot - \sin ( e ^ { x ^ { 5 } \cdot \sin ( x ) } ) \cdot - \sin ( x ) - x ^ { 5 } e ^ { x ^ { 5 } \cdot \sin ( x ) } \cdot \sin ( e ^ { x ^ { 5 } \cdot \sin ( x ) } ) \cos ( x ) } { 2 \sqrt { \cos ( e ^ { x ^ { 5 } \sin ( x ) } ) } } } \end{array} \right. \)

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

(b) \( f(x)=-x^{6}-8 x^{5}-7 x^{4}+6 x^{2} \) Falls to the left and rises to the right Rises to the left and falls to the right Falls to the left and falls to the right

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".

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) \( (- \) infinity, 0\( ) \) (c) \( f \) is concave up on the open interval(s) (-infinity , 0\( ) \) (d) \( f \) is concave down on the open interval(s) \( (0 \), infinity) (e) the \( x \) coordinate(s) of the points of inflection are \( -\sqrt{39}, 0, \sqrt{39} \) 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".