Click on the graph of the original function

Click on the graph of the first derivative

Click on the graph of the second derivative

## About First and Second Derivatives

The graph of a function $$f(x)$$ is closely related to the graphs of its first and second derivatives:
• When the graph of the function $$f(x)$$ is increasing, the value of $$f'(x)$$ is positive, so the graph of $$f'(x)$$ will lie above the $$x$$-axis.
• When the graph of the function $$f(x)$$ is decreasing, the value of $$f'(x)$$ is negative, so the graph of $$f'(x)$$ will lie below the $$x$$-axis.
• When the slope of the line tangent to $$f(x)$$ is 0, the value of $$f'(x)$$ is 0, so the graph of $$f'(x)$$ will have an $$x$$-intercept.
• When the graph of the function $$f(x)$$ is concave up, the value of $$f''(x)$$ is positive, so the graph of $$f''(x)$$ will lie above the $$x$$-axis.
• When the graph of the function $$f(x)$$ is concave down, the value of $$f''(x)$$ is negative, so the graph of $$f''(x)$$ will lie below the $$x$$-axis.
• When the graph of $$f(x)$$ has an inflection point, the value of $$f''(x)$$ is 0, so the graph of $$f''(x)$$ will have an $$x$$-intercept.
• If $$f(x)$$ is a polynomial with degree $$n$$, then $$f'(x)$$ will be a polynomial of degree $$n-1$$ and $$f''(x)$$ will be a polynomial of degree $$n-2$$.
Using these clues, it is possible to determine which of the graphs in this applet is the original function, which is its first derivative, and which is its second derivative.

## Using the Applet

The applet will generate three graphs, representing the functions $$f(x)$$, $$f'(x)$$, and $$f''(x)$$. The function $$f(x)$$ will be a polynomial (that is randomly generated using a selection of points within the graphing window and Lagrange interpolation). By studying the three graphs and the properties above, decide which of the three graphs is the actual function $$f(x)$$, and click on it. You'll then be asked to decide which of the two remaining graphs is $$f'(x)$$.