Question

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

Answer 1

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`