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  2. AP Maths
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Graphs of Functions & Their Derivatives

Sarah Miller

Sarah Miller

5 min read

Next Topic - Second Derivative Test for Local Extrema

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Study Guide Overview

This study guide covers the concept of concavity, including concave up and concave down curves, determined by the sign of the second derivative. It explains how to identify points of inflection where concavity changes. The guide provides exam tips, a worked example, practice questions, a glossary, and real-world applications.

Introduction to Concavity

#Understanding Concavity

Concavity describes the direction in which a curve bends and is related to the second derivative of a function.

Key Concept

Concavity is determined by the sign of the second derivative, f’’(x){f}^{\text{'}\text{'}}(x)f’’(x).

#Concave Up and Concave Down

A curve is:

  • Concave up if f’’(x)≥0{f}^{\text{'}\text{'}}(x) \ge 0f’’(x)≥0 for all values of xxx in an interval.

    • f’(x){f}^{\text{'}}(x)f’(x) is increasing in this interval.
    • For example, the function f(x)=x2f(x) = x^2f(x)=x2 is concave up because f’’(x)=2≥0{f}^{\text{'}\text{'}}(x) = 2 \ge 0f’’(x)=2≥0.

  • Concave down if f’’(x)≤0{f}^{\text{'}\text{'}}(x) \le 0f’’(x)≤0 for all valu...

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Previous Topic - Candidates Test for Global ExtremaNext Topic - Second Derivative Test for Local Extrema

Question 1 of 8

🎉 Concavity of a function is determined by the sign of which derivative?

First derivative

Second derivative

Original function

Third derivative