what does say about f ?

What does say about f ?Increasing/decreasing test

If on an interval I, then f is increasing on I. If on an interval I, then f is decreasing on I.

Proof. Use Lagrange’s mean value theorem.

( ) 0 f x( ) 0 f x

f

2 1 2 1( ) ( ) ( )( )f x f x f c x x

Example Ex. Find where the function is increasing

and where it is decreasing. Sol. Since when x0,and f is not differentiable at 0, we knowf is increasing on (-1,0), (2,+1); decreasing on (0,2).

23( ) ( 5) f x x x

1 2 13 3 32 5( ) ( 5) ( 2),

3 3

f x x x x x x

Example Ex. Find the intervals on which is i

ncreasing or decreasing. Sol. increasing on decreasing on

Ex. Prove that when Sol. Let f(x)=sinx+tanx-2x, x2I =(0,/2). Then

f increasing on I, and f(x)>f(0)=0 on I.

3 2

10( )4 9 6

f xx x x

1( ,1)2 1( ,0), (0, ), (1, )

2

sin tan 2 x x x 0 .2

x

2 2 22

1 1( ) cos sec 2 cos 2 (cos ) 0cos cos

f x x x x xx x

Example Ex. Show that is decreasing on (0,1). Sol.

Ex. Prove that when x>0.

Sol.

ln( )1

x xf xx

2

1 ln( ) 0? 0?(1 )

x xf x

x1( ) 1 ln ( ) 1 0 ( ) (1) 0 g x x x g x g x gx

3

sin6

xx x

3 2

( ) sin ( ) cos 1 , ( ) sin 06 2

x xf x x x f x x f x x x

The first derivative test A critical number may not be a maximum/minimum point. The first derivative test tells us whether a critical number

is a maximum/minimum point or not: If changes from positive to negative at c, then maximum If changes from negative to positive at c, then minimum If does not change sign at c, then no maximum/minimum This explains why has no maximum/minimum at

0.

f f f

3( )f x x

Example Ex. Find all the local maximum and minimum values of

the function Sol.

All critical numbers are:Using the first derivative test, we know: is local maximum point, are local minimum points

2 23( ) ( 1) .f x x

3

4( ) 0 0.3 ( 1)( 1)

xf x xx x

0, 1, 1.

0 1

Convex and concave Definition If the graph of f lies above all of its tangents on

an interval I, then it is called convex (concave upward) on I; if the graph of f lies below all of its tangents on I, it is called concave (concave downward) on I. The property of convex and concave is called convexity (concavity).

Definition A point on the graph of f is called an inflection point if f is continuous and changes its convexity.

The convexity of a function depends on second derivative.

Convexity test If for all x in I, then f is convex on I. If for all x in I, then f is concave on I.

Ex. Find the intervals on which is convex or concave and all inflection points.

Sol.

By convexity test, f convex on andconcave on and the inflection points are

( ) 0f x ( ) 0f x

2

1( )1

xf xx

2

2 3

2( 1)( 4 1)( ) 0 1, 2 3( 1)

x x xf x xx

( 2 3, 2 3) (1, )( , 2 3) ( 2 3,1).

1, 2 3, 2 3.

The second derivative test The second derivative can help determine whether a

critical number is a local maximum or minimum point. The second derivative test If then f has a local minimum at c If then f has a local maximum at c Ex. Find the local maximum and minimum points of

Sol. Local maximum local minimum

( ) 0, ( ) 0,f c f c ( ) 0, ( ) 0,f c f c

1( ) cos cos 2 .2

f x x x

x k2 42 , 23 3

x k x k

Before sketching a graph Using derivative to find the global and local maximum and

minimum values, and locate critical numbers Using derivative to find convexity and locate inflection

points Using derivative to find intervals on which the function is

increasing or decreasing Find domain, intercepts, symmetry, periodicity and

asymptotes

Asymptotes Horizontal asymptotes: if then y=L is a horizontal asymptote of the curve y=f(x)

Vertical asymptotes: if then x=a is a vertical asymptote of the curve y=f(x)

Slant asymptotes: if or

then y=mx+b is a slant asymptote of the curve y=f(x)

lim ( ) or lim ( ) ,

x x

f x L f x L

lim ( ) or lim ( ) ,

x a x a

f x f x

lim[ ( ) ( )] 0x

f x mx b

lim [ ( ) ( )] 0x

f x mx b

Slant asymptote Since to find slant

asymptotes, we first investigate the limit

if it exists, then and y=mx+b is a slant

asymptote.

( )lim[ ( ) ( )] 0 lim ,

x x

f xf x mx b mx

lim ( ) / ,

x

m f x x

lim( ( ) ),

x

b f x mx

Homework 9 Section 4.3: 14, 16, 17, 47, 49, 70, 74