NOTE: This question is fairly sophisticated and is here to show you some of the complications that can occur. If you don't fully understand it now, don't worry!
when `x = 0`, `x = 5`
When `x = 0`, `y = 0`.
3. maxima and minima?
when `x = 0` or `x = 4`
So we have max or min at `(0,0)` and `(4,-256)`.
4. Second derivative:
Now `y” = 0` for `x = 0` so `(0,0)` is ???
Now `y” > 0` for `x = 4` so `(4,-256)` is a local MIN
5. Points of inflection
We now use the second derivative to find points of inflection:
when `x = 0` or `x = 3`
If `x < 0`, `y” < 0`:
If `0 < x < 3`, `y” < 0`:
There is no sign change, so at `x = 0`, there is NO point of inflection.
If `x > 3`, `y” > 0`:
So the sign of `y”` has changed, so `(3,-162)` is a point of inflection.
Actually, at `x = 0`, we have a FLAT SPOT. It is not a local maximum, even though it may appear so.
So we are ready to sketch the curve:
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