Given the graph of f(x), what do you know about the value of f’ and f ” at each indicated point?

The graph of 
1

3
3 x x is given below.
a) What are the x-coordinates of the relative maxima and minima of f(x)?
b) What are the x-coordinates for the point(s) of inflection of f(x)?
c) What is f’(x)? Draw the graph of f’ on the grid. d) What are the x-intercepts of f’ ?
e) What are the x-coordinates where the maximum and minimum values of f’ occur?
f) What is f”(x) ? Draw the graph of f” on the grid above.
g) What is the x-intercept of f”(x)?

What relationships do you notice
(about the extrema and point of inflection of f)?

a) On what interval(s) is f’(x) above the x-axis? b) On what interval(s) is f’(x) below the x-axis?
What connections can you make between the graph of f(x) and whether f’ is above or below the x-axis?

c) On what intervals is f’(x) decreasing?
d) On what intervals is f’(x) increasing?
What connections can you make between the graph of f(x) and whether f’ is increasing or decreasing?

There is a maximum on the graph of f(x) when the graph of f’(x)
There is a minimum on the graph of f(x) when the graph of f’(x) …

There is a point of inflection on the graph of f(x) when f’(x)…
The graph of f(x) is concave down when the graph of f’(x) is …

The graph of y = h ( x ) is shown. Which of the following could be the graph of y = h ′( x ) ?

The graph of y = f’ ( x ) is shown. Which of the following could be the graph of y = f ( x ) ?
A.
B.
f ’ (x)
C.
D.

The figure above shows the graph of f’, the derivative of a function f. The domain is the set of all real numbers x such that -10 < x < 10.
(a) For what values of x does the graph of f have a horizontal tangent line?
(b) For what values of x in the interval -10 < x < 10 does the graph of f have a relative maximum? Explain.
(c) For what value(s) of x does f have a point of inflection?
(d) For what value(s) of x is the graph of f concave downward? Justify your answer.