Chapter 5 § 3 Maxima and Minima

Terminology Maximum/Minimum Value b/Point (a,b) f ’ (x) changes from +ve to -ve / from -ve to +ve f ” (x) 0 Turning Point :f ’ (x) changes sign through the point. Critical Point or Stationary Point:f ’ (x) = 0 Absolute Maximum/Minimum Value b /Point(a,b) b > f(x)/b < f(x) Increasing/Decreasing function f ’ (x) > 0/f ’ (x) < 0

Quiz graphically information

Monotonic functions : Increasing function Increasing function is a function f satisfying: f(x 1 )  f(x 2 ) for x 1 <x 2 How to show that a function is increasing? 1. Direct proof 2. f ’ (x)  0 Give an inequality. f(a)  f(x)  f(b) a b y=f(x)

Monotonic functions : Decreasing function Decreasing function is a function f satisfying: f(x 1 )  f(x 2 ) for x 1 <x 2 How to show that a function is increasing? 1. Direct proof 2. f ’ (x)  0 Give an inequality. f(a)  f(x)  f(b) a b y=f(x)

Application:Proving Inequalities Example 4.4 If x>0, prove that x-x 2 /2<ln(1+x)<x-x 2 /2+x 3 /3 1.Construct a function Let f(x) = ln(1+x)-x+x 2 /2. 2. Is it monotonic? 3. Are there any extreme values? 4.Construct the inequality. 5.Construct another function for the other.

Illustrative Examples Example 4.5 Show that for 0<x<  /2, (a)x<tanx(b)x+x 3 /3<tanx (c) 2x/  < sinx Example 4.3 Read yourself

§5&6 Absolute Maximum Value What is the absolute maximum point? (c, f(c)) How to determine without graph? By differentiation. f ’ (x)  0 for x  [a, c] & f ’ (x)  0 for x  [c, b] Give an inequality. f(c)  f(x) for all x  [a, b] a b y=f(x) c

§5&6 Absolute Minimum Value a b y=f(x) c What is the absolute minimum point? (c, f(c)) How to determine without graph? By differentiation. f ’ (x)  0 for x  [a, c] & f ’ (x)  0 for x  [c, b] Give an inequality. f(c)  f(x) for all x  [a, b]

Application on inequalities E.g.6.2 Show that x-lnx  1 for all x>0, and that the equal sign holds if and only if x=1.[Graph][Graph] Proof: 1.Formulate the function Let f(x) = x-lnx [or f(x) = x-lnx-1] 2.Investigate f ’ (x) f ’ (x) 0 for x>1. 3.Find the least value. f(1)=1 [or f(1)=0] 4.Construct the inequality. f(x)  f(1)=1 [f(x)  f(1)=0] Classwork Ex.5.5 Q.2

§8 Convexity of functions x1x1 x2x2 x A 1 (x 1,y 1 ) A 2 (x 2,y 2 ) A B If A 1 B:BA 2 =p:q and p+q=1, find the coordinates of A and B in terms of x i ’ s, y i ’ s, p and q. Construct an inequality involving f, p, q and x 1 and x 2.

Convexity of a function A function is convex on an interval I if it satisfies that f(px 1 +qx 2 )  pf(x 1 )+qf(x 2 ), for any x 1, x 2  I and p+q=1 concave 

9.Point of Inflexion Definition: A point of inflexion or a point of inflection or inflexional point or inflectional point (x 0,y 0 ) on a curve y=f(x) is a continuous point at which the function f(x) changes from convex to concave.

How many inflexional points are there? x x<-11<x<1 x>1 y ” x x< 0-1<x<1 x>1 y ”

10. Asymptotes to a Curve

How many kinds of asymptotes are there? A) Horizontal asymptote y=a B) Vertical asymptote x=b C) Oblique asymptote y= ax + b

Definition of an asymptote to a curve A straight line is an asymptote to a curve if and only if the perpendicular distance form a variable point on the curve to the line approaches to zero as a limit when the point tends to infinity along the curve on both sides or one side of the curve.

Theorem 10.1 y=ax+b is an oblique asymptote to y=f(x) iff Proof

Thm 10.2 Proof

Examples of finding asymptotes Example 10.2 Example 10.1, Example 10.1 Supp.eg10.6 Find the oblique asymptote of y = (x 2 -x+1) 0.5 Supp.eg10.6 Example10.5 Find the oblique asymptote of y=(2x 3 -x 2 +3x+1)/(x 2 +1) Example

Oblique asymptote for special rational function f(x)=P(x)/Q(x), where deg(P(x)) = deg(Q(x))+1 The oblique asymptote is y= the quotient when P(x) is divided by Q(x)

Procedures of sketching curves Procedures of sketching curves e.g. Domain of f(x) Symmetry of f(x):periodicity, even or odd Find f ’ (x) and f ” (x) Monotonicity and Extrema of f(x) Convexity and Inflexional points of f(x) Asymptotes Special points e.g. intercepts

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