Answer to Question #130941 in Calculus for Morgan

Question

let f(x)=(6/x^2 +6). find all intervals where f is concave up and all intervals where it is concave down.

Answer 1

A function is concave up for the intervals where d²f(x)/dx² >0

from f(x) =6/x² +6, df(x)/dx = -12x^-3 or df(x)/dx =-12/x³,

d²f(x)/dx² = 36x^-4 or d²f(x)/dx² = 36/x⁴ +0 >0

12/x⁴ >0 is true.

A function f(x)=6/x²+6 is concave up for every $x \neq0.$

A function is concave down for the intervals where d²f(x)/dx² <0

from f(x) =6/x² +6, df(x)/dx = -12x^-3 or df(x)/dx =-12/x³

d²f(x)/dx² = 36x^-4 or d²f(x)/dx² = 36/x⁴ +0 <0

12/x⁴ cannot be negative, hence the function f(x) is not concave down.