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:

• 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 \(n-2\).