Analysis of a Function

 

 

• Critical number:  Value of x where  or   fails to exist.  (Remember, the x must be in the domain of f to be called a “critical number”.
• If a function has relative extrema, they must occur at critical numbers.
• Just because a function has critical numbers, does not necessarily imply that it has relative extrema.  You must perform a first or second derivative test.

• The Extreme Value Theorem and how to use it.

§      If f is continuous on [a,b], then f has both an absolute maximum and an absolute minimum function value.

• The Mean Value Theorem

§      If f is continuous on [a,b] and differentiable on (a,b) then there exists a  such that .

§      It states that if a function is well-behaved, on any particular interval, there is a place where the instantaneous rate of change in function values is equal to the average rate of change in function values.

• If
• If

• The First Derivative Test for Relative extrema.

§      If  is a critical number and  before a and  after a, then f has a relative minimum of  at .

§      If  is a critical number and  before a and  after a, then f has a relative maximum of  at .

• The Second Derivative Test for Relative extrema

§      If  is a critical number and  then the curve is concave up and f has a relative minimum at .

§      If  is a critical number and  then the curve is concave down and f has a relative maximum at .

§      Note: If  is zero or nonexistent, you cannot use the 2^nd derivative test

• If
• If
• Values of x where  are possible inflection points

§      If f has an inflection point, it must occur at a place where

§      Just because  does NOT mean there is an inflection point—you must demonstrate that the second derivative changes sign!