Concavity of Polynomial Functions

Explore interactively the concavity of polynomial functions.

The concavities of the graphs of polynomial functions of the form

f(x) = x 3 + a x 2 + bx + c

are explored, using the graph of f, the tangent to the graph of f and the graph of the second derivative.

Interactive Tutorial

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1 - Click on the button "click here to start" and maximize the window obtained. Three graphs are shown: in blue color the graph of the polynomial function f. In red color the tangent line to the graph of f and in black color the graph of the second derivative f '' which is drawn as the position of the tangent line is changed using the bottom slider ("Change Tangent Position").

2 - Use the top slider to set coefficient a to 3, coefficient b to 2 and coefficient c to zero so that function f has the following formula:

f(x) = x 3 + 3 x 2 + 2 x


2 - Use the slider at the bottom of the left panel to change the position of the tangent line (red). As you start from the left moving to the right the slope of the tangent line(red) decreases. What is the sign of f ''? Is the graph (blue color) concave up or down?

At x = 1, the slope changes from decreasing to increasing. The point (1 , f(1)) on the graph is called an inflection point. For x greater than 1, what is the sign of f "? Is the graph (blue) concave up or down?

3 - You will now use analytical methods based on the theorems on concavity to explain the observations you made.

4 - 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?

5 - You can now set coefficients a, b and c to different values, study the concavity graphically and then compare the graphical observations to your analytical results.

6 - You can also use the above applet to study and understand points of inflections. These are points where the concavity of a graph changes.