Understanding Concavity and Inflection Points in Mathematics

Chabot Mathematics §3.2 Concavity& Inflection Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

3.1 Review § • Any QUESTIONS About • §3.1 → Relative Extrema • Any QUESTIONS About HomeWork • §3.1 → HW-13

§3.2 Learning Goals • Introduce Concavity (a.k.a. Curvature) • Use the sign of the second derivative to find intervals of concavity • Locate and examine inflection points • Apply the second derivatives test for relative extrema

ConCavity Described • Concavity quantifies the Slope-Value Trend (Sign & Magnitude) of a fcn when moving Left→Right on the fcn Graph m≈0 m≈−1.4 m≈−4.4 m≈+2.2 m≈+2.2 m≈−4.4 m≈−1.4

MATLAB Code % Bruce Mayer, PE % MTH-15 •11Jul133 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The data blue =[2.2 0 -1.4 -4.4] red = [-4.4 -1.4 0 2.2] % % the 6x6 Plot axes; set(gca,'FontSize',12); subplot(1,2,1) bar(blue, 'b'), grid, xlabel('\fontsize{14}Position, x'), ylabel('\fontsize{14}m = df/dx'),... title(['\fontsize{16}MTH15 • BLUE',]), axis([0 5 -5,3]) subplot(1,2,2) bar(red, 'r'), grid, xlabel('\fontsize{14}Position, x'), axis([0 5 -5,3]),... title(['\fontsize{16}MTH15 • RED',]) set(get(gco,'BaseLine'),'LineWidth',4,'LineStyle',':')

ConCavity Defined • A differentiable function f on a < x < b is said to be: … concave DOWN (↓) if df/dx is DEcreasingon the interval …concave up if df/dx is INcreasingon the interval.

Example  Graphical Concavity • Consider the function f given in the graph and defined on the interval (−4,4). • Approximate all intervals on which the function is INcreasing, DEcreasing, concave up, or concave down

Example  Graphical Concavity • SOLUTION • Because we have NO equation for the function, we need to use our best judgment: • around where the graph changes directions (increasing/decreasing) • where the derivative of the graph changes directions (concave up or down).

Example  Graphical Concavity • To determine where the function is INcreasing, we look for the graph to “Rise to the Right (RR)” Rising

Example  Graphical Concavity • Similarly, the function is DEcreasing where the graph “Falls to the Right (FR)”: Falling

Example  Graphical Concavity • Conclude that f is increasing on the interval (0,4) and decreasing on the interval (−4,0) • Now ExamineConcavity. Falling to Rt Rising to Rt

Example  Graphical Concavity • A function is concave UP wherever its derivative is INcreasing. Visually, we look for where the graph is“curved upward”, or “Bowl-Shaped”Similarly, A function is concave DOWN wherever its derivative is DEcreasing. Visually, we look for where the graph is “curved downward”, or “Dome-Shaped”

Example  Graphical Concavity • The graph is “curved UPward” for values of x near zero, and might guess the curvature to be positive between −1 & 1 f is ConCave UP

Example  Graphical Concavity • The graph is “curved DOWNward” for values of x on the outer edges of the domain. f is ConCave DOWN f is ConCave DOWN

Example  Graphical Concavity • Thus the function is concave UP approximately on the interval (−1,1) and concave DOWN on the intervals (−4, −1) & (1,4) f is ConCave DOWN f is ConCave DOWN f is ConCave UP

Inflection Point Defined • A function has an inflection pointat x=a if f is continuous and the CONCAVITY of f CHANGES at Pt-a ConCave UP InflectionPoint ConCave DOWN

MATLAB Code % Bruce Mayer, PE % MTH-15 • 10Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = -2; xmax = 9; ymin =-50; ymax = 50; % The FUNCTION x = linspace(xmin,xmax,1000); y =(x-4).^3/4 + (x+5).^2/7; yOf4 = (4-4).^3/4 + (4+5).^2/7 % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, 'LineWidth', 5),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),... title(['\fontsize{16}MTH15 • Inflection Point',]) hold on plot(4, yOf4, 'd r', 'MarkerSize', 9,'MarkerFaceColor', 'r', 'LineWidth', 2) plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:10:ymax]) hold off

Example  Inflection Graphically change from concave down to up change from concave up to down • The function shown above has TWO inflection points.

2nd Derivative Test • Consider a function for Which is Defined on some interval containing a critical Point (Recall that ) Then: • If , then is Concave UP at so is a Relative MIN • If , then is Concave DOWN at so is a Relative MAX

Example  Apply 2ndDeriv Test • Use the 2nd Derivative Test to Find and classify all critical points for the Function • SOLUTION • Find the critical points by solving:

Example  Apply 2ndDeriv Test • By Zero-Products: • Also need to check for values of x that make the derivative undefined. • ReCall the 1st Derivative: • Thus df/dx is UNdefined for x = −1, But the ORIGINAL function is ALSO Undefined at the this value • Thus there is NO Critical Point at x = −1

Example  Apply 2ndDeriv Test • Thus the only critical points are at −2 & 0 • Now use the second derivative test to determine whether each is a MAXimum or MINimum (or if the test is InConclusive):

Example  Apply 2ndDeriv Test • Before expanding the BiNomials, note that the numerator and denominator can be simplified by removing a common factor of (x+1) from all terms:

Example  Apply 2ndDeriv Test • Now expand BiNomials: • Now Check Value of f’’’(0) & f’’’(−2)

Example  Apply 2ndDeriv Test • The 2nd Derivative is NEGATIVE at x = −2 • Thus the orginalfcn is ConCaveDOWN at x = −2, and aRelative MAX exists at this Pt • Conversely, 2nd Derivative is POSITIVE at x = 0 • Thus the orginalfcn is ConCave UP at x= 0 anda Relative MIN exists at this Pt

Example  Apply 2ndDeriv Test • Confirm by Plot → • Note the relative MINimum at 0, relative MAXimumat −2, and a vertical asymptote where the function is undefined at x=−1 (although the vertical line is not part of the graph of the function)

ConCavity Sign Chart • A form of the df/dx (Slope) Sign Chart (Direction-Diagram) Analysis Can be Applied to d2f/dx2 (ConCavity) • Call the ConCavity Sign-Charts “Dome-Diagrams” for INFLECTION Analysis ConCavityForm ++++++ −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points a b c Inflection NOInflection Inflection

Example  Dome-Diagram • Find All Inflection Points for • Notes on this (and all other) PolyNomial Function exists for ALL x • Use the ENGR25 Computer Algebra System, MuPAD, to find • Derivatives • Critical Points

Example  Dome-Diagram • The Derivatives • The Critical Points • The ConCavity Values Between Break Pts • At x = −1 • At x = ½ • At x = ½

MyPAD Code

Example  Dome-Diagram • Draw Dome-Diagram • The ConCavity Does NOT change at 0, but it DOES at 1 • Since Inflection requires Change, the only Inflection-Pt occurs at x = 1 ConCavityForm −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points 0 1 NOInflection Inflection

Example  Dome-Diagram • TheFcnPlotShowingInflectionPoint at(1,y(1))= (1,−3) (1,−3)

MATLAB Code % Bruce Mayer, PE % MTH-15 • 11Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = -1.5; xmax = 2.5; ymin =-15; ymax = 15; % The FUNCTION x = linspace(xmin,xmax,1000); y =3*x.^5 - 5*x.^4 - 1; % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, 'LineWidth', 5),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 3x^5 - 5x^4 - 1'),... title(['\fontsize{16}MTH15 • Dome-Diagram',]) hold on plot(1,-3, 'd r', 'MarkerSize', 10,'MarkerFaceColor', 'r', 'LineWidth', 2) plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:5:ymax]) hold off

Example  Population Growth • A population model finds that the number of people, P, living in a city, in kPeople, t years after the beginning of 2010 will be: • Questions • In what year will the population be decreasing most rapidly? • What will be the population at that time?

Example  Population Growth • SOLUTION: • “Decreasing most rapidly” is a phrase that requires some examination. “Decreasing” suggests a negative derivative. • “Decreasing most rapidly” means a value for which the negative derivative is as negative as possible. In other words, where the derivative is a MIN

Example  Population Growth • Need to find relative minima of functions (derivative functions are no exception) where the rate of change is equal to 0. • “Rate of change in the population derivative, set equal to zero” TRANSLATES mathematically to

Example  Population Growth • The only time at which the second derivative of P is equal to zero is the beginning of 2013. • Need to verify that the derivative is, in fact, negative at that point: 

Example  Population Growth • Thus the function is decreasing most rapidly at the inflection point at the beginning of 2013: • The Model Predicts 2013 Population: • x

WhiteBoard Work • Problems From §3.2 • P45 → Sketch Graph using General Description • P66 → Spreading a Rumor

All Done for Today RememgeringConCavity:cUP & frOWN

Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

ConCavity Sign Chart ConCavityForm ++++++ −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points a b c Inflection NOInflection Inflection

Max/Min Sign Chart Slope ++++++ −−−−−− −−−−−− ++++++ df/dx Sign x Critical (Break)Points a b c Max NOMax/Min Min

Understanding Concavity and Inflection Points in Mathematics