1. 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.

2. Example
Ex. Find where the function is increasing and where it is decreasing.
Sol. Since when x¹0,
and f is not differentiable at 0, we know
f is increasing on (-1,0), (2,+1); decreasing on (0,2).

3. Example Ex. Find the intervals on which is increasing or decreasing.
Sol. increasing on
decreasing on
Ex. Prove that when
Sol. Let f(x)=sinx+tanx-2x, x2I =(0,p/2). Then
f increasing on I, and f(x)>f(0)=0 on I.

4. Example Ex. Show that is decreasing on (0,1). Sol.
Ex. Prove that when x>0.

5. 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.

6. 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

7. 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.

8. 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 and
concave on and the inflection points
are

9. 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

10. 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

11. 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)

12. Slant asymptote Since to find slant
asymptotes, we first investigate the limit
if it exists, then and y=mx+b is a slant
asymptote.

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