From parts (a), (b), and (c) we know the following about
:
- The critical numbers of
(where
is zero or does not exist) are ![{\displaystyle \displaystyle x=0,1}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/7002e1d4d77379ccdf456df57b79e6db7afc7263)
is increasing on
and decreasing on ![{\displaystyle x\in (-\infty ,0)\cup (1,\infty )}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/72e23fc504e5b719879fa645d91f84514d6992ee)
has a local minimum at
and a local maximum at
(note the change in the increase/decrease of
)
is concave up on
and concave down on ![{\displaystyle x\in (-{\tfrac {5}{2}},0)\cup (0,\infty )}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/65aeb5836af5d3647e5140ad2c4a020032a8cad6)
has an inflection point (changes concavity) at ![{\displaystyle x=-{\tfrac {5}{2}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/953855a80ebf2d4374e8360d26411110af0eebf4)
and thus
has x-intercepts at ![{\displaystyle x=0,{\tfrac {9}{2}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/6551aa6991c686f2a303425ae7d18555c76ef493)
and
, thus the graph of
is very steep near ![{\displaystyle x=0}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc)
Plotting the points of interest gives the following:
Accounting for increase and decrease gives the following rough sketch:
Finally, incorporating concavity leads us to the final graph:
![{\displaystyle \displaystyle f(x)=63x^{2/7}-14x^{9/7}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/abf08d4e8b37e67b5aeb2abfbeed7ec9c530a52a)