Increase and Decrease of Functions

1. Definition of Increase and Decrease
   For a function , given any two numbers and in an interval:
1. If and , the function is said to be increasing on that interval.
2. If and , the function is said to be decreasing on that interval.


2. Determining Increase and Decrease Using Derivatives
   If a function is differentiable on an open interval, then:
1. If for all in that interval, the function is increasing.
2. If for all in that interval, the function is decreasing.

1. Increasing, then .
2. Decreasing, then .

Maximum and Minimum of Functions

1. Definition of Local Maximum and Minimum
   For a function at a point :
1. If for all in some open interval containing , then has a local maximum at , and is the maximum value.
2. If for all in some open interval containing , then has a local minimum at , and is the minimum value.

2. If a function is continuous at and:
1. Increasing then decreasing around , it has a local maximum at .
2. Decreasing then increasing around , it has a local minimum at .


3. Using Derivatives to Determine Maxima and Minima
   For a differentiable function at , if :
1. If the sign of changes from positive to negative around , has a local maximum at .
2. If the sign of changes from negative to positive around , has a local minimum at .


4. If has a local extremum at and is differentiable at that point, then . However, a function may still have an extremum even if does not exist (e.g., at ).
   
   
5. Using the Second Derivative for Maxima and Minima
   For a function with a second derivative, if
1. If , has a local maximum at .
2. If , has a local minimum at .


6. If both and , it is inconclusive whether has an extremum at . For example, has neither maximum nor minimum at , while has a minimum at .

Concavity and Convexity of Curves

1. For two points and on the curve :
1. If the curve lies below the line segment joining and , the curve is concave down (or convex up) in that interval.
   
   
2. If the curve lies above the line segment joining and , the curve is concave up (or convex down) in that interval.
   
   
2. Using the Second Derivative to Determine Concavity
   For a function with a second derivative:
1. If , the curve is concave up (or convex down) in that interval.
2. If , the curve is concave down (or convex up) in that interval.
   
   

Inflection Points

1. Definition of an Inflection Point
   A point on the curve is called an inflection point if the curve changes from concave up to concave down, or vice versa, at .
   
   
2. Determining Inflection Points
   For a function with a second derivative, if and the sign of changes around , then is an inflection point.
   
   
3. If , it does not guarantee an inflection point. For example, has , but there is no inflection point at .

Sketching the Graph of a Function

1. The domain and range of the function.
2. Symmetry and periodicity.
3. Intercepts with the axes.
4. Points of increase, decrease, maxima, and minima.
5. Concavity, convexity, and inflection points.
6. Asymptotes.

1. If or , the line is a vertical asymptote.
2. If or , the line is a horizontal asymptote.

Global Maximum and Minimum