How do you determine whether the function #f(x)= (x-1) / (x+52)# is concave up or concave down and its intervals?

1 Answer
Sep 17, 2015

Answer:

Use the sign of the second derivative (or knowledge of transformations of the reciprocal function).

Explanation:

Calculus

Using calculus, the general method of determining concavity is to investigate the sign of the second derivative.

#f(x)= (x-1) / (x+52)#

#f'(x)= 53 / (x+52)^2#

#f''(x)= -106 / (x+52)^3#

For this function, the sign of #f''# is the opposite of the sign of #x+52#.

#f''# is positive on the interval #(-oo,-52)# and negative on #(-52,oo)#.

So the graph of #f# is concave up interval #(-oo,-52)# and concave down on #(-52,oo)#.

Because #-52# is not in the domain of #f#, there is no inflection point.
(The definition of inflection point that I am accustomed to is: a point on the graph at which the concavity changes.)

Reciprocal Function

#f(x)= (x-1) / (x+52)# can be written:

#f(x)= ((x+52)-53) / (x+52) = (x+52)/(x+52) -53/(x+52) = 1-53/(x+52)#

From the graph of #y = 1/x#

graph{y=1/x [-20.28, 20.27, -10.14, 10.14]}

we obtain the graph of #f# by
translating #52# left, expanding vertically by a factor of #53#, reflect in the #x# axis, and the translate up #1# unit.

graph{y=(x-1)/(x+52) [-123.7, 42.94, -35.4, 48]}

Because of the reflection the graph is concave up on the left and concave down on the right. The horizontal translation moves the change in concavity from #x=0# to #x=-52#