A function \(f(x)\) is said to be *increasing* on an interval \(I\) if, for all \(x_0\) and \(x_1\) in \(I\) with \(x_0\lt x_1\), we have \(f(x_0) \lt f(x_1)\). This means that as you trace the graph from left to right on such an interval, your finger would move up. Similarly, a function is *decreasing* on an interval \(I\) if, for all \(x_0\) and \(x_1\) in \(I\) with \(x_0\lt x_1\), we have \(f(x_0) \gt f(x_1)\).

A function \(f(x)\) is said to be*concave up* on an interval \(I\) if its first derivative is increasing on \(I\). Graphically, this means the function is curved and forming a bowl shape. Similarly, a function is *concave down* when its first derivative is decreasing. When a function is concave down, it is curved and forming an upside down bowl (open umbrella) shape.

The graph of a function \(f(x)\) is closely related to the graphs of its first and second derivatives:

A function \(f(x)\) is said to be

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 \(n-2\).

The applet will generate the graph of a random polynomial function. Use the controls to highlight the intervals of the graph where it is increasing, decreasing, concave up, and concave down, and to display the first and second derivatives.

This applet was created using JavaScript and the Raphael library. If you are unable to see the applet, make sure you have JavaScript enabled in your browser. This applet may not be supported by older browsers.