1. Найдите наименьшее значение функции y = -2 ln(-5x - 9)^5 - 50x - 5 на промежутке [-2.8; -1.9].

Answer

To find the minimum value of the function y = -2 ln(-5x - 9)^5 - 50x - 5 on the interval [-2.8; -1.9], we first simplify the expression using logarithm properties: y = -10 ln(-5x - 9) - 50x - 5. Next, we find the derivative of the function with respect to x: y' = -10 * (1 / (-5x - 9)) * (-5) - 50 = 50 / (-5x - 9) - 50 = 50(1 / (-5x - 9) - 1). Setting the derivative to zero to find critical points: 1 / (-5x - 9) = 1 => -5x - 9 = 1 => -5x = 10 => x = -2. Check if x = -2 lies in the interval [-2.8; -1.9], which it does. Now, evaluate the function at the critical point and the endpoints of the interval. For x = -2: y(-2) = -10 ln(-5(-2) - 9) - 50(-2) - 5 = -10 ln(1) + 100 - 5 = 0 + 95 = 95. For the endpoints, if we approach x = -1.9, -5(-1.9) - 9 = 9.5 - 9 = 0.5, y(-1.9) = -10 ln(0.5) - 50(-1.9) - 5 = 10 ln(2) + 95, which is greater than 95. Similarly, for x = -2.8, -5(-2.8) - 9 = 14 - 9 = 5, y(-2.8) = -10 ln(5) + 140 - 5 = 135 - 10 ln(5), which is also greater than 95. Thus, the minimum value on the given interval is 95.