Al calcular la segunda derivada de la función \( y=\csc x \) se obtiene: \( \begin{array}{l}\frac{\partial^{2} y}{\partial x^{2}} \\ \text { a. } \frac{\partial^{2} y}{\partial x^{2}} \\ \text { at } x+\csc ^{2} x \\ \text { c. } \frac{\partial^{2} y}{\partial x^{2}}\end{array}=\csc x\left(\cot ^{2} x+\csc ^{2} x\right) \) d. \( \frac{\partial^{2} y}{\partial x^{2}}=\csc x(\cot x+\csc x) \)

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".

(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) \( \lim _{x \rightarrow \infty}\left(\frac{x-1}{x+3}\right)^{x+2} \)

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

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

(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