# Analyze the graph of a derivative

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. :)

## 1 Answer

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.

• Well, using your suggestion, i believe that the critical values are: x=0, x=1 and x=2. There's a relative extreme on x = 1. And the interval where the second derivative is negative: (2,0). Am i wrong? – egarro Oct 15 '14 at 16:18