The basic shape of a cubic is:
Keeping this in mind helps with the sketching process.
Now `y=0` when
`x = 0`, `x = -3` and `x = 3`
When `x = 0`, `y = 0`.
3. maxima and minima?
`x=-sqrt(3)` or `x=sqrt(3)`
So we have max or min at approximately `(-1.7,10.4)` and `(1.7,-10.4)`.
[We could check which is which by trying some points near `-1.7` and `+1.7` to determine what the sign changes are. But we need to find the second derivative anyway for points of inflection, so we use that to determine max or min.]
4. Second derivative:
If `x = -1.7`, `y” < 0`, so MAX at `(-1.7,10.4)`
If `x = +1.7`, `y” > 0`, so MIN at `(1.7,-10.4)`
5. Point of inflection:
`(d^2y)/(dx^2)=0` when `x = 0`
changes sign from negative (concave down) to positive (concave up) as x passes through `0`.
So we are ready to sketch the curve:
Graph of `y=x^3-9x`.
The following points are indicated with dots:
Local maximum (-1.7,10.4)
Point of inflection (0,0)
Local minimum (1.7,-10.4)