Fermat’s Theorem If f has a local maximum or minimum at c, and if  exists, then .

A critical number of a function f is a number c in the domain of fsuch that either  or  does not exist.

If f has a local extremum at c, then c is a critical number of f.

Increasing/Decreasing Test

• If  on an interval, then f is increasing on that interval.
• If 

The First Derivative Test Suppose that c is a critical number of a continuous function f.

• If  changes from positive to negative at c, then f has a local maximum at c.
• If  changes from negative to positive at c, then f has a local minimum at c.
• If  does not change sign at c, (that is, 

A function (or its graph) is called concave upward on an interval I if  is an increasing function on I. It is called concave downward on I if  is decreasing on I.

Concavity Test

• If  for all x in I, then the graph of f is concave upward on I.
• If 

EXAMPLE: Find the critical numbers of .
 

The domain of  is , that is, all real numbers except —2. [Note: If the index of the radical had been even, I would have had to exclude any value that made the radicand negative (in this case, any value of  less than 4). However, since the index is odd (5, in this case), I do not have to worry about the radical. The fifth root of a negative number is a real number.]

is undefined if  or if .

The critical numbers of  are 0 and 4. [Note: -2 is not a critical number since —2 is not in the domain of .]
 
 

Use the first derivative test to find where the local maximum(s) and/or local minimum(s) of  occur. Find the exact coordinates of the local extrema.

Table 1: 
 

Interval	Sign of	Graphical Behavior of
(-*,-2)	Positive	Increasing
(-2,4)	Positive	Increasing
(4,13)	Positive	Increasing
(13,*)	Negative	Decreasing

[Note: Even though —2 isn’t a critical number of , we need to include it in the table because the graphical behavior of  might change at a discontinuity of .]

According to the First Derivative Test and Table 1,  has a local maximum at . The exact coordinates of the local maximum are 
 
 
 
 

EXAMPLE: Find the critical numbers of .

The domain of  is (0,*).

is undefined on (-*, 0).

if 

The only critical number of  is .

EXAMPLE: Find the critical numbers of  on [0, 21).

is never undefined

if  or 

Therefore, the critical numbers of  on [0, 21) are 0, 1/2, 1, or 31/2.

EXAMPLE: Suppose that . Where is  increasing? decreasing?

is undefined

if 

Table 2: 

Interval	Sign of	Graphical behavior of
(-*, 0)	Negative	Decreasing
(0,1)	Negative	Decreasing
(1,*)	Positive	Increasing

According to Table 2, the function  is increasing on (1,°) and decreasing on (-°, 1).
 
 
 
 

Where is  concave up? concave down?

is undefined

if 

Table 3: 

Interval	Sign of	Graphical behavior of
(-*,-1)	Positive	Concave up
(-1,0)	Negative	Concave down
(0,1)	Negative	Concave down
(1,*)	Positive	Concave up

According to Table 3, the function  is concave up on (-*,-1) and (1,*) and concave down on .