Applications of Derivatives

⭐ Important Questions

1. Find intervals where f(x) = x³ - 3x is increasing/decreasing.
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   f'(x) = 3x² - 3 = 3(x-1)(x+1)\nCritical: x = -1, 1\nx 0 => increasing\n-1 < x < 1: f'(x) decreasing\nx > 1: f'(x) > 0 => increasing
2. A wire of length 28 m is cut into two pieces for squares. Find lengths to minimize total area.
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   Let pieces: 4x and 28-4x, sides: x and 7-x\nArea A = x² + (7-x)² = 2x² - 14x + 49\nA'(x) = 4x - 14 = 0 => x = 3.5\nA''(x) = 4 > 0 => minimum\nLengths: 14 m and 14 m