Homepage

Course Index

Build a Quiz

Subscriptions

Courses

NSW Y12 Maths Standard 1 NSW Y12 Maths Standard 2 NSW Y12 Maths Advanced NSW Y12 Maths Extension 1 NSW Y12 Maths Extension 2

Topics

Sequences and Series Graphs and Equations Differential Calculus Integration Exponential and Log Function Trig Functions Motion and Rates Series and Finance Descriptive Statistics Bivariant Data Analysis Random Variables

NSW Y12 Maths - Advanced Differential Calculus Concavity and Points of Inflexion

Resources for Concavity and Points of Inflexion

  • Questions

    11

    With Worked Solution
    Click Here
  • Video Tutorials

    1


    Click Here
  • HSC Questions

    2

    With Worked Solution
    Click Here

Concavity and Points of Inflexion Theory

\textbf{Concavity:}\\ If \(f^{\prime \prime}(x)>0\) the curve is concave upwards.\\ If \(f^{\prime \prime}(x)<0\), the curve is concave downwards.\\ \textbf{Points of inflexion}\\ If \(f^{\prime \prime}(x)=0\), and concavity changes, it is a point of inflexion.\\ If \(f^{\prime}(x)=0\) also, it is a horizontal point of inflexion.\\ \begin{multicols}{2} \textbf{Example 1}\\ %51959 For the curve \(y=x^3-6x^2+5x-2\), determine the values of \(x\) for which the curve is concave up.\\ \textbf{Example 1 solution}\\ \(\begin{aligned} y&=x^3-6 x^2+5 x-2 \\ y^{\prime}&=3 x^2-12 x+5 \\ y^{\prime \prime}&=6 x-12 \end{aligned}\)\\ \(\begin{aligned} \text { Concave up} &\rightarrow y^{\prime \prime}>0 \end{aligned}\)\\ \(\begin{aligned} \therefore 6 x-12&>0 \\ x&>2 \end{aligned}\)\\ \columnbreak \textbf{Example 2}\\%10274 The points of inflection of the curve \(y = {x^4} - 6{x^2} + 2x - 4\) are?\\ \textbf{Example 2 solution}\\ \(\begin{aligned} y&=x^4-6x^2+2x-4\\ y'&=4x^3-12x+2\\ y''&=12x^2-12 \end{aligned}\)\\ \(\begin{aligned} \text{Point of inflection occur}&\text{when } y''=0 \end{aligned}\)\\ \(\begin{aligned} 12x^2-12&=0\\ x^2-1&=0\\ (x-1)(x+1)&=0\\ x=1 \text{ or } x&=-1 \end{aligned}\)\\ \(\begin{aligned} \text{At } x=1,\quad y=1-6+&2-4=-7 \rightarrow (1,-7)\\ \text{At } x=-1,\, y=1-6-&2-4=-11 \rightarrow (-1,-11) \end{aligned}\)\\ \(\text{Testing }(1,-7)\)\\ \(\begin{array}{r} \text{At }x=0.9 ,\;\; y''<0\\ \text{At }x=1.1 ,\;\; y''>0\\ \end{array}\)\(\begin{array}{r} \text{Concavity change }\end{array}\)\\ \(\therefore \text{ point of inflection}\)\\ \(\text{Testing }(-1,-11)\)\\ \(\begin{array}{r}\text{At }x=-1.1 ,\;\; y''>0\\\text{At }x=-0.9 ,\;\; y''<0\\\end{array} \begin{array}{r} \text{Concavity change } \end{array}\)\\ \(\therefore \text{ point of inflection}\) \end{multicols}

Create account

I am..

Please enter your details

I agree with your terms of service




Videos

Videos relating to Concavity and Points of Inflexion.

  • Concavity and Points of Inflexion - Video - Concavity, Inflection Points and Second Derivatives

Plans & Pricing

With all subscriptions, you will receive the below benefits and unlock all answers and fully worked solutions.

  • Teachers Tutors
    Features
    Free
    Pro
    All Content
    All courses, all topics
     
    Questions
     
    Answers
     
    Worked Solutions
    System
    Your own personal portal
     
    Quizbuilder
     
    Class Results
     
    Student Results
    Exam Revision
    Revision by Topic
     
    Practise Exams
     
    Answers
     
    Worked Solutions
  • Awesome Students
    Features
    Free
    Pro
    Content
    Any course, any topic
     
    Questions
     
    Answers
     
    Worked Solutions
    System
    Your own personal portal
     
    Basic Results
     
    Analytics
     
    Study Recommendations
    Exam Revision
    Revision by Topic
     
    Practise Exams
     
    Answers
     
    Worked Solutions