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

Question

Emilia Aguilar · Accepted Answer

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

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#