Introducing Calculus
Investigation

Talking about curves
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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?

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Based on this NRICH resource, used with permission. The NRICH resource remains Copyright University of Cambridge, All rights reserved.

 Last updated 08-Jul-16
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