Reconstruct f  from its Second Derivative Function

 HELP




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.

Explore

  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?


This page is part of the GeoGebra Calculus Applets project.