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# Inflection points review

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.4 (EK)
,
FUN‑4.A.5 (EK)
,
FUN‑4.A.6 (EK)
Review your knowledge of inflection points and how we use differential calculus to find them.

## What are inflection points?

Inflection points (or points of inflection) are points where the graph of a function changes concavity (from \cup to \cap or vice versa).

## Practice set 1: Analyzing inflection points graphically

Problem 1.1
How many inflection points does the graph of f have?

Want to try more problems like this? Check out this exercise.

## Practice set 2: Analyzing inflection points algebraically

Inflection points are found in a way similar to how we find extremum points. However, instead of looking for points where the derivative changes its sign, we are looking for points where the second derivative changes its sign.
Let's find, for example, the inflection points of f, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, 2, end fraction, x, start superscript, 4, end superscript, plus, x, cubed, minus, 6, x, squared.
The second derivative of f is f, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesis, equals, 6, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, plus, 2, right parenthesis.
f, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesis, equals, 0 for x, equals, minus, 2, comma, 1, and it's defined everywhere. x, equals, minus, 2 and x, equals, 1 divide the number line into three intervals:
Let's evaluate f, start superscript, prime, prime, end superscript at each interval to see if it's positive or negative on that interval.
Intervalx-valuef, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesisVerdict
x, is less than, minus, 2x, equals, minus, 3f, start superscript, prime, prime, end superscript, left parenthesis, minus, 3, right parenthesis, equals, 24, is greater than, 0f is concave up \cup
minus, 2, is less than, x, is less than, 1x, equals, 0f, start superscript, prime, prime, end superscript, left parenthesis, 0, right parenthesis, equals, minus, 12, is less than, 0f is concave down \cap
x, is greater than, 1x, equals, 2f, start superscript, prime, prime, end superscript, left parenthesis, 2, right parenthesis, equals, 24, is greater than, 0f is concave up \cup
We can see that the graph of f changes concavity at both x, equals, minus, 2 and x, equals, 1, so f has inflection points at both of those x-values.
Problem 2.1
g, left parenthesis, x, right parenthesis, equals, x, start superscript, 4, end superscript, plus, 4, x, cubed, minus, 18, x, squared
For what values of x does the graph of g have a point of inflection?