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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/ 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(x1) f(x2) for x1<x2
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(x1) f(x2) for x1<x2
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-x2/2<ln(1+x)<x-x2/2+x3/31. Construct a function
Let f(x) = ln(1+x)-x+x2/2.2. Is it monotonic?3. Are there any extreme values?4. Construct the inequality.5. Construct another function for the other.
Illustrative Examples
Example 4.5Show that for 0<x</2,(a) x<tanx (b) x+x3/3<tanx(c) 2x/ < sinx
Example 4.3Read 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] Proof:1. Formulate the function
Let f(x) = x-lnx [or f(x) = x-lnx-1]2. Investigate f ’(x)
f ’(x)<0 for 0<x<1 and 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
Classwork Ex.5.5 Q.2
§8 Convexity of functions
x1
x2x
A1(x1,y
1)
A2(x2,y2)
A
BIf A1B:BA2=p:q and p+q=1, find the coordinates of A and B in terms of xi’s , yi’s, p and q.Construct an inequality involving f, p, q and x1 and x2.
Convexity of a function A function is convex on
an interval I if it satisfies that
f(px1+qx2) pf(x1)+qf(x2),
for any x1, x2I and p+q=1
-1.0 1.0 2.0 3.0
1.0
2.0
x
y
concave
9. Point of Inflexion Definition: A point of inflexion or a
point of inflection or inflexional point or inflectional point (x0,y0) 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<-1 1<x<1
x>1
y” + - +
x x< 0
-1<x<1 x>1
y” - - +
1.
2
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
aylimx
ylimbx
x)x(f
limax
]ax)x(f[limbx
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 10.1y=ax+b is an oblique asymptote to y=f(x) iff 0]bax)x(f[lim
x
Proof
Thm 10.2
Proof
]ax)x(f[limb and x
)x(flima
xx
Examples of finding asymptotes Example 10.2 Example 10.1, Supp.eg10.6 Find the oblique
asymptote of y = (x2-x+1)0.5
Example10.5 Find the oblique asymptote of y=(2x3-x2+3x+1)/(x2+1)
10.4
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 curvese.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
31
32
)x6(xy
Sketch of 31
32
)x6(xy
