Warm Up
Complete the worksheet
Relationships between graphs and their derivatives
Graph Analysis
Day 1
1 3
The graph of f ( x )  x  x is given below.
3
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?
Make some conjectures…
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?
Summary
F(x)
F’(x)
F”(x)
Summarize…
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) …
Summarize…
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) ?
f ’ (x)
The graph of y = f’(x) is shown. Which of the
following could be the graph of y = f(x) ?
A.
C.
B.
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.