Sketch the graph of f showing the correct concavity with the points of inflection properly labelled.

Question

Given the graph of the derivative of the function f, which is continuous in its domain, which is the set of real numbers.

Hello! Please help me understand how to solve this problem :(( I would really appreciate it if there will be enough justifications for the response. Thank you in advance!!

Answer 1

If the graph shown is f'(x), then there are 2 points of inflection where f'(x) has a maximum and a minimum, i.e. f"(x) has a zero. The point of inflection on the right has a horizontal tangent. Presumably this is the graph of a

rational function with a zero in the denominator where x=1. The best way I can help you with the "concavity" question is to tell you that on the left the function graph looks like a cubic polynomial and on the right it looks like y=(1-x)³. You should make a sketch.