Finding Local Maximum, Local Minimum, and Inflection Points
To find a
local maximum
(point A on the graph of f(x), where f(x) changes from increasing to
decreasing), we can find root A on the graph of f '(x), where f '(x)
crosses the x-axis from positive to negative. To find a
local minimum
(point C on the graph of f(x), where f(x) changes from decreasing to
increasing), we can find root C on the graph of f '(x), where f '(x)
crosses the x-axis from negative to positive.
To find root A on your graph of f '(x), from the GRAPH
menu, press MORE, then MATH (F1), then ROOT (F1). If you have more
than one function graphed, use the up or down arrow to move the cursor
to the graph of f '(x). At the prompt "Left Bound?", trace a bit to
the left of root A, then ENTER. At the prompt "Right Bound?", trace a
bit to the right of root A, then ENTER. At the prompt "Guess?" trace
to somewhere near root A. Then press ENTER, and wait for the
calculator to find the root. Follow the same procedure to find root C.
Alternatively, you can find the local maximum and local minimum
directly, on the graph of f(x). To find the local maximum, from the
GRAPH menu, press MORE, then MATH (F1), then FMAX (F5). If you have more
than one function graphed, use the up or down arrow to move the cursor
to the graph of f(x). At the prompt "Left Bound?", trace a bit to the
left of your local maximum, then ENTER. Then trace a bit to the right
of your local maximum and ENTER your right bound. Then trace to
somewhere near the local maximum and ENTER your guess. Then wait for
the calculator to find the local maximum. Follow the same
procedure to find a local minimum, only use FMIN (F4) on the GRAPH
MATH menu.
To find an
inflection point (point B on the graph of f(x), where f(x) changes from concave
down to concave up (or vice versa, depending on your graph)) you can
either (a) use MORE MATH ROOT to find where the second derivative
crosses the x-axis; or (b) use MORE MATH and FMIN or FMAX to find the
corresponding local minimum or local maximum on the graph of the first derivative;
or (c) use MORE MATH MORE INFLC to find the inflection point directly,
on the graph of f(x). INFLC will prompt you for a left bound, right
bound, and guess.
When using FMIN, FMAX, or INFLC with more than one function
graphed, be sure that the cursor is on the correct function. When
prompted for the left bound, right bound, and guess, you can trace
with the left/right arrow keys or type in a number. After the guess,
press ENTER and wait for the result.
To find the absolute maximum and
absolute minimum, that is, the absolute
highest and lowest points on your entire graph, first find all local
maximum and minimum points. Then find the y-values at the beginning
and end of your graph (easiest way is to use TRACE and type in the
x-value), and compare them with your local maximum and minimum values.
The absolute maximum and absolute minimum might be at local maximum or
local minimum points, or at the beginning or end of the graph.
If your function is complicated (for example, if you have
a logistic model), an easy way to graph the first and second
derivative without having to type them all in is to do the following.
Enter your original function in y1. Then in y2, press 2nd CALC (it's
on the divide key), then select der1 (F3) for first derivative. Type
in the name of your function, y1, a comma, then x, then close the
parentheses. It should look like:
y2=der1(y1,x)
Similarly, in y3, select der2 (F4) from the CALC menu, and type in
y1, a comma, then x, then close the parentheses. It should look like:
y3=der2(y1,x)
This tells the calculator to graph the first and second
derivatives of the function you have in y1. (The "x" specifies the
name of the variable.)
Page written and maintained by
Helen Read.
