Reconstruct `f` from its Second Derivative Function


We previously addressed the question, "Given `f'(x)`, what can we know about `f(x)`?" But now we go deeper:
Given the graph of `f''(x)` what can we deduce about `f(x)`?
You are given the graph of `f''(x)`, and your task is to reconstruct the graph of `f(x)`. Recall some important information...
  • If `f''(x) > 0`, then `f'(x)` is increasing, so `f(x)` is concave up.
  • If `f''(x) < 0`, then `f'(x)` is decreasing, so `f(x)` is concave down.


  1. The graph of `f''(x)` is shown in purple. Drag the blue points up and down so that together they follow the shape of the graph of `f(x)`. As a help, the three large green points are points on the graph of `f(x)`.
  2. How can you tell where the `f`-graph has inflection points?
  3. Just from looking at the graph of `f''(x)`, how easy or difficult is it to tell `f(x)` is increasing or decreasing?
  4. What information in the `f''`-graph would tell you the point where `f` increases the fastest?
  5. When `f''(c)` is positive and `f''(c)` is a local max for `f''`, then `f(x)` is "maximally" concave up at `c`. What does this look like in the `f` graph?