Study and do approximate graphic presentation of the following function. The study should contain:
Roots and infinities
Increasing and decreasing intervals
Stationary points and their characteristics
Points of infinite derivative and their characteristics
Inflection points
Parts
There is one root: x=0 .
There are two points of infinity: x=±1 .
At x=−1
the infinity is positive at the left
and negative at the right.
At x=+1
the infinity is positive at the left
and negative at the right as well.
The derivative is
From the derivative (part 4) it follows that
there are no stationary points and
the infinite derivatives belong to the infinite points of the function.
The second derivative is
which vanishes at the point x=0 .
In order to check if x=0 is an inflection point, we need the third derivative there
which obviously does not vanish there (=12) and therefore x=0 is an inflection point.
The function is increasing with two points of discontinuity (infinities) at x=±1 .
Remark
The function is odd and therefore the derivative is even as expected.
Score
Parts 6 and 7 are worth two points each.
The rest (parts 1,2,3,4,5,8) are worth 1 point each.