The function is .
The domain of the function is all real numbers.
\Intercepts:
\Find the -intercept by substituting
in
.
The -intercept is
.
Find the -intercept by substituting
in
.
The -intercept is
.
Find the extrema of .
Differentiate on each side with respect to .
.
Find the critical numbers by solving .
.
or
The critical points are ,
.
Relative extrema points exist at critical numbers.
\Substitute in the function.
.
Substitute in the function.
.
Maximum is at and minimum at
.
Consider the test intervals as ,
and
.
\
Interval \ | \
Test Value | \Sign of ![]() | \
Conclusion | \
![]() | \
![]() | \
\
| \
Decreasing | \
![]() | \
![]() | \
\
| \
Increasing | \
![]() | \
![]() | \
\
| \
Decreasing | \
Find the inflection points.
\
Differentiate on each side with respect to .
.
.
Equate to
.
.
,
, and
.
The inflection points occurs at ,
and
.
Substitute in the function.
.
Substitute in the function.
.
Substitute in the function.
.
Inflection points are ,
and
.
Consider the test intervals as ,
,
and
.
\
Interval \ | \
Test Value | \Sign of ![]() | \
Concavity | \
![]() | \
![]() | \
\
| \
Down | \
![]() | \
![]() | \
\
| \
Up | \
![]() | \
![]() | \
\
| \
Down | \
![]() | \
![]() | \
\
| \
Up | \
Find asymptote of function .
Find the horizontal asymptote by evaluating .
Take common numerator and denominator by .
The horizontal asymptote is .
Find the vertical asymptotes by equating denominator to zero.
\Since the equation has no real solutions, there are no vertical asymptote.
\Graph the function .
The domain of the function is all real numbers.
\Intercept is .
There is no vertical asymptote.
\The horizontal asymptote is .
Maximum is at and minimum at
.
Inflection points are ,
and
.
Graph:
\Draw the coordinate plane.
\Plot the intercepts, asymptote, maximum, minimum and inflection points.
\Connect the curve with plotted points.
\Graph of the function :
Graph of the function :
.