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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 5

Lesson 7: Determining concavity of intervals and finding points of inflection: algebraic

# Concavity review

Review your knowledge of concavity of functions and how we use differential calculus to analyze it.

## What is concavity?

Concavity relates to the rate of change of a function's derivative. A function f is concave up (or upwards) where the derivative f, prime is increasing. This is equivalent to the derivative of f, prime, which is f, start superscript, prime, prime, end superscript, being positive. Similarly, f is concave down (or downwards) where the derivative f, prime is decreasing (or equivalently, f, start superscript, prime, prime, end superscript is negative).
Graphically, a graph that's concave up has a cup shape, \cup, and a graph that's concave down has a cap shape, \cap.

## Practice set 1: Analyzing concavity graphically

Problem 1.1
• Current
Select all the intervals where f, prime, left parenthesis, x, right parenthesis, is greater than, 0 and f, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesis, is greater than, 0.

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

## Practice set 2: Analyzing concavity algebraically

Problem 2.1
• Current
f, left parenthesis, x, right parenthesis, equals, 3, x, start superscript, 4, end superscript, minus, 16, x, cubed, plus, 24, x, squared, plus, 48
On which intervals is the graph of f concave down?