IB›IB Maths: Analysis and Approaches HL›Mind maps5.2 Increasing and decreasing functionsIB Maths: Analysis and Approaches HL: Mind mapStudy pack PDFAlso for this subtopic:Revision notesFlashcardsSubtopic testCover factsThe testf′(x)>0f'(x)>0f′(x)>0: increasingf′(x)<0f'(x)<0f′(x)<0: decreasingf′(x)=0f'(x)=0f′(x)=0: horizontal tangentAnswer with intervals of xxx, not yyyMethod Differentiate to get f′(x)f'(x)f′(x) Solve f′(x)=0f'(x)=0f′(x)=0 for critical values Test the sign on each interval State the intervals Worked examplef(x)=x3−12x+1f(x)=x^3-12x+1f(x)=x3−12x+1f′(x)=3(x−2)(x+2)f'(x)=3(x-2)(x+2)f′(x)=3(x−2)(x+2)Positive quadratic is negative between its rootsDecreasing for −2<x<2-2<x<2−2<x<2Rising, fallingsign of the derivativef′>0f'>0f′>0Graph of f′Above the axis: fff increasingBelow the axis: fff decreasingCrosses the axis: fff turnsTouches the axis: no turnIncreasing everywhereShow f′(x)>0f'(x)>0f′(x)>0 for all xxxQuadratic f′f'f′: negative discriminantAlways increasing means one-to-oneExam tips3x2−12=03x^2-12=03x2−12=0 gives x=±x=\pmx=± 2, not 4f′(3)=0f'(3)=0f′(3)=0 is not f(3)=0f(3)=0f(3)=0Interpret with sign, size and unitsClosed interval: compare endpoints too