Question

How do you determine whether the function `f(x)=3/(x^2−4)` is concave up or concave down and its intervals?

Answer 1

Explained below.

Explanation:

f '(x)= `(-6x)/(x^2 -4)^2`. Critical points are that at which f '(x) =0 or where it does not exist. These in this case are -2,0,2. Hence the interval to be tested are `(-oo, -2) , (-2,0), (0,2) and (2,oo)`

For using second derivative test, f " (x)=`6(3x^2+4)/(x^2−4)^3`

Select any test value in these intervals and test for the sign of f "(x).
In `(oo,-2)` it would be >0, conclusion, concave up
In`(-2,0)`, it would be <0, " " , concave down
In `(0,2)`, it would be <0, " " , concave down
In `(2,oo)`, it would be >0 " " , concave up