How derivatives Affect the shape of a graph

Question

4. T/F (with justification) If a function f(x) on the interval (−1,1) is twice differentiable and f′′(c) = 0 for some c in (−1,1) then f(x) has an inflection point at x = c.

Answer 1

f " = 0 only tells you POTENTIAL points of inflection. You must always check to see if there is a change in concavity at that point. If concavity changes, then it's a POI. If not, it is not a POI.

Answer 2

False

the function is twice differentiable on the interval (-1,1) so the function is smooth and continuous. This is an important distinction because if you have some function with breaks in it or sharp points then the following is not necessarily true.

We know that f''(c) = 0 would be a critical point however for it to be an inflection point the sign of f''(x) would have to change value as it crosses c and this is not always the case.

Take the function f(x) = x⁴

f''(x) = 12x²

f''(0) = 0 however f''(x) ≥ 0 over the region (-1,1) so this function would not have an inflection point at x=0 and the concavity remains positive over (-1,1).