Chapter 5: Graphs and derivative

5.3. Higher Derivatives, Second Derivative Test

Higher derivatives: We know that denotes the(first) derivative of a function f(x).  The derivative of will be denoted by .  (This is also called the second derivative of f (x)).  Next, the derivative of will be denoted by .  (This is also called the third derivative of f(x)).  The fourth derivative of f(x) will be denoted by .  In fact the notation for higher derivatives of f(x) is given by , where n is any integer greater than 3.

Example: Let f(x) = .  Then =5(this is of course the derivative of f, according to the power rule),  (this is obtained by computing the derivative of );   ( this is obtained by computing the derivative of ;  = 60 (this is the derivative of );  ( the derivative of ).  In fact for any integer n greater than 4.

Concavity:  Let f be a function with derivatives and existing at all point in an interval (a,b).  Then f is concave upward on (a, b) provided that for all x in (a, b).

f is concave downward  on (a, b) if for all x in (a, b).

Notice that if f is concave upward on an interval (a, b) then there should be a local minimum in the interval.  Likewise if f is concave downward on (a, b) then the interval admits a local maximum.  This is the content of second derivative test.

Second Derivative Test:

Let exists on some open interval containing a critical number c ( that is ).
1.  If then f assumes its minimum at c and that f(c) is the minimum value of f
2. If then f assumes its maximum at c and that f(c) is the maximum value of f
3. If , the test gives no information about the extrema

Example:
Let f(x) =  - - 10 x - 25.  Then .  Therefore, by setting the derivative to zero and solve it for x,  x = -5 is the critical point.  Furthermore, since for any x, then .  That is .  Hence the critical point, x = - 5, is the place where f assumes its maximal value.  The maximal value of f is given by f (-5) =-25-(-50)-25 = 0