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ACMMM103

5.7 Maximum Rates of Increase or Decrease

Maximum Rates of Increase or Decrease

  • Earlier, we noticed how the first derivative f^{\prime}(x) can be used to find maximum and minimum values.
  • Similarly, we can find maximum and minimum values for f^{\prime}(x) by using its derivative, f^{{\prime}{\prime}}(x) i.e. the second derivative.
  • Recall that f^{\prime}(x) is interpreted as the rate of increase/decrease/change, therefore f^{{\prime}{\prime}}(x) is capable of finding the maximum and minimum rate of increase or decrease.
  • A table is constructed below to illustrate the cases.
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5.6 Maximum and Minimum in Real Life

Maximum and Minimum in Real Life

  • In real life, many things can be maximised or minimised (e.g. cost, area, volume, profit, sales, etc.) for various purposes (e.g. in manufacturing, marketing etc.).
  • Using differentiation and our knowledge in absolute maximum and minimum can help to solve these problems.

Example

A farmer has sufficient fencing to make a rectangular pen of perimeter 200 metres. What dimensions will give an enclosure of maximum area?

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