Taking it further

What are the different possibilities of sign combination for a curve which turns exactly twice?

How can you be sure that you have found them all?

  • How do zeros of the function and zeros of the gradient enter into the analysis?

  • Are there any curves which always have the same sign for the function and gradient?

  • Can you find algebraic expressions for any of the functions you have used in this problem?

  • How does the number of different regions vary for curves with other numbers of turns?

  • Is it possible to draw a curve for which there is only one region?