I'm not sure why you substituted in the second derivative.
But substituting in gives points (1,1/2) and (-1,-1/2)
We know these points are either local or global maxima/minima.
We only have to check whether there exist such that or
(if you check the graph of y(x) on wolframalpha we actually see that (1,1/2) and (-1,-1/2) are global maxima/minima. Thus I don't understand why your
text states (-1,-1/2) is a local minimum. )