Page 51 - Calc for the Clueless

Definition

Inflection point—a point at which a curve goes from up to down or down to up.

A test for inflection points is f''(c) = 0. There are in fact two tests to tell whether a point is an inflection point.

Test 1 (Easier)

Note
If the third derivative is very difficult to find, use the other test.

A. If , then c is an inflection point.

B. If f'"(c) = 0, this test fails and you must use another test.
Test 2

-
A. If f"(c ) and f"(c ) have different signs, c is an inflection point.
+
+
B. If f"(c ) and f"(c ) have the same sign, c is not an inflection point.
-
There is also a second and easier test for max and min points. Suppose f'(c) = 0. If f"(c) is positive, it means the
slope is increasing. This means the curve is facing up, which means a minimum. Suppose f'(c) = 0 and f"(c) is
negative. This means the slope is decreasing, the curve faces down, and we have a maximum. If f"(c) = 0, then
we use the other test.
Problem

Before we sketch some more curves, let's make sure we all understand each other. There is a kind of problem
my fellow lecturer Dan Mosenkis at CCNY likes to give his students. It's not my cup of tea or cup of anything
else, but I think it will help you a lot. We have made up a craaaazy function, f(x). Its picture is on the next page.
For each listed value of x, A through K, look at f(x) and estimate the sign of f(x), f'(x), and f"(x) at each point.
Enter one of the following symbols on the chart: + (if positive), - (if negative), 0, and ? (if it does not exist).