Find the x-coordinates of the points at which the function is at a maximum or a minimum and indicate which (maximum or minimum, relative or absolute).
I know that two of those points are $x = -2$ and $x = 2$ but my book states that $x=-3$ is also a relative minimum.
However, I don't understand why. My book states that:
In an open (non-empty) interval the relative extremes can happen:
- at the zeros of the derivative function where there is a sign change
- at points of the interval where there is no derivative and the lateral derivatives either have different signs of one of them is zero
Following thay logic, shouldn't $x=3$ be a relative maximum? The sign of the derivative is negative on the left and positive on the right.
And why is $x=-3$ considered a relative minimum? It's derivative is unknown, it's left side derivative is non-existant and the only thing I know about it is that the right side derivative is negative.
To sum up, my questions are:
- Shouldn't $x=3$ be a relative maximum? And why not?
- Why is $x=-3$ considered a relative minimum?