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

1 Answer
bp
Jan 6, 2016

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

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