How do you determine whether the function #f(x)= (x-1) / (x+52)# is concave up or concave down and its intervals?
1 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.
For this function, the sign of
So the graph of
Because
(The definition of inflection point that I am accustomed to is: a point on the graph at which the concavity changes.)
Reciprocal Function
From the graph of
graph{y=1/x [-20.28, 20.27, -10.14, 10.14]}
we obtain the graph of
translating
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