Question

1. the function ( f ) and its first and second derivatives are given:

( f(x)=\frac{x}{x^{2}-1}, quad f^{prime}(x)=\frac{-left(x^{2}+1
ight)}{left(x^{2}-1
ight)^{2}}, quad f^{prime prime}(x)=\frac{2 xleft(x^{2}+3
ight)}{left(x^{2}-1
ight)^{3}} ),
obtain the following information. write none when appropriate.
(a) ( x )-intercepts:________, ( y )-intercepts:________.
(b) vertical asymptotes: ________. horizontal asymptotes: ________.
(c) increasing on: ________, decreasing on: ________.
(d) coordinates of local maxima: ________, local minima: ________.
(e) concave upwards on: ________, downwards on: ________.
(f) coordinates of inflection points: __________.
(g) use this information to sketch the graph of ( f )

Explanation:

Step1: Find x - intercepts

Set \(y = f(x)=0\), so \(\frac{x}{x^{2}-1}=0\). Since \(x^{2}-1
eq0\) (denominator), then \(x = 0\).

Step2: Find y - intercepts

Set \(x = 0\), then \(y=f(0)=\frac{0}{0^{2}-1}=0\).

Step3: Find vertical asymptotes

Set \(x^{2}-1 = 0\), so \(x=\pm1\).

Step4: Find horizontal asymptotes

\(\lim_{x
ightarrow\pm\infty}f(x)=\lim_{x
ightarrow\pm\infty}\frac{x}{x^{2}-1}=\lim_{x
ightarrow\pm\infty}\frac{\frac{x}{x^{2}}}{\frac{x^{2}}{x^{2}}-\frac{1}{x^{2}}}=0\)

Step5: Determine increasing/decreasing intervals

\(f^{\prime}(x)=\frac{-(x^{2}+1)}{(x^{2}-1)^{2}}\). Since \(x^{2}+1>0\) and \((x^{2}-1)^{2}>0\) for \(x
eq\pm1\), \(f^{\prime}(x)<0\) for \(x\in(-\infty,-1)\cup(-1,1)\cup(1,\infty)\). So the function is decreasing on \((-\infty,-1)\), \((-1,1)\) and \((1,\infty)\) and there are no increasing intervals.

Step6: Find local maxima/minima

Since \(f^{\prime}(x)\) never changes sign (always negative except at \(x=\pm1\) where it is undefined), there are no local maxima or minima.

Step7: Determine concavity

Set \(f^{\prime\prime}(x)=0\), \(\frac{2x(x^{2}+3)}{(x^{2}-1)^{3}}=0\). Since \(x^{2}+3>0\), then \(x = 0\).
Test intervals:
For \(x\in(-\infty,-1)\), let \(x=-2\), \(f^{\prime\prime}(-2)=\frac{2(-2)((-2)^{2}+3)}{((-2)^{2}-1)^{3}}=\frac{-2\times7}{27}<0\)
For \(x\in(-1,0)\), let \(x =-\frac{1}{2}\), \(f^{\prime\prime}(-\frac{1}{2})=\frac{2(-\frac{1}{2})((-\frac{1}{2})^{2}+3)}{((-\frac{1}{2})^{2}-1)^{3}}=\frac{- \frac{13}{2}}{(-\frac{3}{4})^{3}}>0\)
For \(x\in(0,1)\), let \(x=\frac{1}{2}\), \(f^{\prime\prime}(\frac{1}{2})=\frac{2(\frac{1}{2})((\frac{1}{2})^{2}+3)}{((\frac{1}{2})^{2}-1)^{3}}=\frac{\frac{13}{2}}{(-\frac{3}{4})^{3}}<0\)
For \(x\in(1,\infty)\), let \(x = 2\), \(f^{\prime\prime}(2)=\frac{2\times2(2^{2}+3)}{(2^{2}-1)^{3}}=\frac{4\times7}{27}>0\)

So the function is concave upwards on \((-1,0)\) and \((1,\infty)\), concave downwards on \((-\infty,-1)\) and \((0,1)\)

Step8: Find inflection points

Set \(x = 0\), \(y=f(0)=0\). So the inflection point is \((0,0)\)

Answer:

(a) \(x\) - intercepts: \(0\), \(y\) - intercepts: \(0\)
(b) Vertical asymptotes: \(x = 1,x=-1\). Horizontal asymptotes: \(y = 0\)
(c) Increasing on: NONE, decreasing on: \((-\infty,-1)\cup(-1,1)\cup(1,\infty)\)
(d) Coordinates of local maxima: NONE, local minima: NONE
(e) Concave upwards on: \((-1,0)\cup(1,\infty)\), downwards on: \((-\infty,-1)\cup(0,1)\)
(f) Coordinates of inflection points: \((0,0)\)