Question

Determine the intervals of concavity and inflection points:
i) f(x) = x^3 - 2X
ii) g(x) = x/(x+1)

Answer 1

Find the second derivative. Where it is positive, it is concave up; negative, concave down. The zeros of the second derivative are points of inflection.

Answer 2

Note that in problem b, the change happens at a point where the function is undefined; this is probably not considered a point of inflection.

Answer 3

if f''(x)=0 undefined, meaning no point of inflection?
if the sign of f''(x) doesnt change for both sides, does it still consider as point of inflection?

Answer 4

\[f(x) = x^3 - 2x \implies f"(x)=6x \implies x=0~inflection~point\]Since f" is negative for x<0, it is concave down in that area, concave up for x>0.