Derivatives, Slope/Tangent Graphic Question

This week I learned in class that the derivative gives us the slope of the tangent line at any point on the graph of f(x). I now need to know how to explain what is happening graphically to the function, f, between x = 6 and x = 8 if f'(6) = 2 and f'(8) = -3.

I am not sure exactly how to explain this.

Re: Derivatives, Slope/Tangent Graphic Question

You know that the graphy of f(x) is trending upward with slopw 2 at x = 6, and it trends downward with slope -3 at x=8. So if the function is continuous that means either (a) there is a point some place between x=6 and x= 8 where the slope is equal to 0, or (b) the function is "jagged" and takes a sharp turn someplace between x=6 and x=8. In either case the function must hit a local maximum in the interval betwen x=6 and x=8.

I think you would find it helpful to sketch some example of graphs that have these slopes, and you'll see how if the slope goes from positive to negative there must be a local maximum in between.