CalcRef

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.

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.

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