IB›IB Maths: Analysis and Approaches SL›Mind maps5.2 Increasing and decreasing functionsIB Maths: Analysis and Approaches SL: Mind mapStudy pack PDFAlso for this subtopic:Revision notesFlashcardsSubtopic testCover factsThe rulef′(x)>0f'(x)>0f′(x)>0: increasingf′(x)<0f'(x)<0f′(x)<0: decreasingf′(x)=0f'(x)=0f′(x)=0: horizontal tangentGraph rises left to right when increasingMethod 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 of xxx 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)Negative between roots: decreasing for −2<x<2-2<x<2−2<x<2Increasing for x<−2x<-2x<−2 or x>x>x> 2Increasingand decreasingf′>0f'>0f′>0f′<0f'<0f′<0Graph of f′f′f'f′ above the axis: fff increasingf′f'f′ below the axis: fff decreasingf′f'f′ crosses the axis: fff turnsf′f'f′ only touches it: no turnAlways increasingShow f′(x)>0f'(x)>0f′(x)>0 for all xxxQuadratic f′f'f′: negative discriminant3x2−6x+k3x^{2}-6x+k3x2−6x+k: 36−12k<036-12k<036−12k<0 gives k>k>k> 3Always increasing means one-to-oneExam tipsAnswer with xxx intervals, not yyy-values3x2−12=03x^{2}-12=03x2−12=0 gives x=±x=\pmx=± 2f′(3)=0f'(3)=0f′(3)=0 is not f(3)=0f(3)=0f(3)=0In context, give sign, size and units