President's Problems

In this issue, we introduce a new feature called the President's Problems. Give them a try. If you wish, you may send solutions to Jim Hall at Parkland. His address is in the mailing list earlier in this issue. (Extra Credit: submit solutions by e-mail using an attached file).

1. The graph of the cubic function f(x) = k(x - a)(x - b)(x - c) has three distinct x-intercepts. Examine the tangent line to the curve at the point . Make a conjecture regarding this tangent line and prove your conjecture.

2. A parabola defined by y = -(x - a)(x - b) is used to define a rectangle whose base is on the x-axis with end points at (a, 0) and (b, 0). the side of the rectangle parallel to this base passes through the vertex of the parabola. What is the relationship between the area of the rectangle and the area between the parabola and the x-axis?

3. Submit a construction for trisecting an angle.

HINT: Reference the IMACC Allerton talk by Underwood Dudley.