For a function such as f(x) = -x^2 +2x +2, a parabola with a maximum at (1,3) a textbook says that the function is increasing in the interval (-infinity, 1]. If this is correct could someone tell me why? I think at x = 1 the slope is 0, so the function is not increasing and the interval should be: (-infinity, 1). Thanks.
If you look at just the interval , you see that for x in that interval but not equal to 1, so f is "increasing at x= 1". Of course, if you use an interval on both sides of x= 1, that is not true.
Of course you could also say "f is decreasing on " so it seems that we are saying f is both increasing and decreasing at x= 1. However, the point is that we are looking at the two intervals separately.