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Applications of Derivatives · Applied Problems

Optimization

Find critical points of f′(x)=0, then verify max/min via endpoint or derivative test\text{Find critical points of } f'(x) = 0 \text{, then verify max/min via endpoint or derivative test}

Optimization finds the maximum or minimum value of a function. Set up the objective function, find critical points, and test them.

  • 2 worked examples
  • 15 in Applications of Derivatives

Worked examples

Find two positive numbers whose sum is 20 and whose product is maximum.
  1. Let x + y = 20, maximize P = xy = x(20-x) = 20x - x²
  2. P'(x) = 20 - 2x = 0 → x = 10
  3. P''(x) = -2 < 0 → local max ✓

Answer: x = y = 10, maximum product = 100.

1 more worked examplePremium

A rectangular pen is built against a straight wall using 100 m of fence on the other three sides. Find the dimensions that maximize its area.

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Common questions

What is the Optimization?

Optimization finds the maximum or minimum value of a function. Set up the objective function, find critical points, and test them. It is one of the applications of derivatives formulas in the CalcRef reference.

How do you use the Optimization?

Worked example. Find two positive numbers whose sum is 20 and whose product is maximum. Let x + y = 20, maximize P = xy = x(20-x) = 20x - x². P'(x) = 20 - 2x = 0 → x = 10. P''(x) = -2 < 0 → local max ✓. Answer: x = y = 10, maximum product = 100.

Practice

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