I have the graph of the derivative of some function:
And i need to know:
a) The critic values of f.
b) The X coordinate of each of points where theres an relative extrema of f.
c) An interval of the domain where the second derivative is negative.
I suppose that the critic values are only x=1, because is where the derivative is cero, the part b i can't get it, and i believe that an interval where the second derivative is negative could be (-2,0), but i'm not really sure. I would appreciate if somebody can help me. :)
a) Critical values are places where the derivative is zero or where the derivative fails to exist. Your graph shows three such points.
b) A relative extreme is a point where the graph changes from increasing to decreasing, or vice versa. In terms of the derivative, it is a place where the derivative switches from positive to negative or vice versa (i.e., where the graph of the derivative crosses the $x$-axis).
c) If the second derivative is negative on an interval, the first derivative is decreasing on that interval. Examine the graph directly to see where this happens.