Question

Знайдіть найменше значення функції на проміжку [-4;4]

у = 10 + 9х - 3х^2 - x^3

Answer 1

$y = - {x}^{3} - 3 {x}^{2} + 9x + 10 \\ y' = - 3 {x}^{2} - 2 \times 3x + 9 = - 3 {x}^{2} - 6x + 9 = \\ = -3 ( {x}^{2} + 2x - 3) = - 3(x + 3)(x - 1) \\ - - - - [ - 3] + + + + [1] - - - - \\ x_{min} = - 3\\ x_{max} = 1\\ \\ y( - 4) = - ( - 4) {}^{3} - 3 \times ( - 4) {}^{2} + 9 \times ( - 4) + 10 = \\ = - ( - 64) - 3 \times 16 - 36 + 10 = 64 - 48 - 26 = - 10 \\ \\ y( - 3) = - ( - 3) {}^{3} - 3 \times ( - 3) {}^{2} + 9 \times ( - 3) + 10 = \\ = - ( - 27) - 3 \times 9 - 27 + 10 = 27 - 27 - 27 + 10 = - 17 \\ \\ y(1) = - {1}^{3} - 3 \times {1}^{2} + 9 \times 1 + 10 = \\ = - 1 - 3 + 9 + 10 = 15 \\ \\ y(4) = - {4}^{3} - 3 \times {4}^{2} + 9 \times 4 + 10 = \\ = - 64 - 3 \times 16 + 36 + 10 = - 18 - 48 = - 66$

Ответ: y min = - 66