# Concavity of Polynomial Functions

The concavity of the graph of the polynomial functions f given by

f(x) = x 3 / 6 - x / 2
is explored, using the graph of f, the first derivative, the tangent to the graph of f and the graph of the second derivative.

## Interactive Tutorial Using the HTML 5 Applet Below

The applet below may be used to visualize the graph of f, the first derivative, the tangent to the graph of f and the graph of the second derivative.
1 - Use the slider at the bottom of the applet to change the position of the tangent line (black). As you start from the left (say from x = - 3) moving to the right up to x = 0, the slope of the tangent line decreases and so does the first derivative (red).
What is the sign of f '' (green) on that interval? Is the graph of f (blue) concave up or down?
2 - At x = 0, the slope of the tangent changes from decreasing to increasing. The point (0 , f(0)) on the graph is called an inflection point where the second derivative change sign and therefore the concavity also changes. For x greater than 0, what is the sign of f "? Is the graph (blue) concave up or down?
Function First Derivative Second Derivative
3 - You will now use analytical methods based on the theorems on concavity to explain the observations you made.
Find the first and second derivatives and then analyze the sign of the second derivative f '' and use the theorems on
concavity . Does the observations you made above and the analytical results agree?