判断函数单调性 - (x/50)^(100/(50-x)) 详细解析与步骤

要判断函数的单调性，需要求出函数的导数。\n\n首先，对函数进行化简：\n\nf(x) = (x/50)^(100/(50-x))\n\n将分子和分母的指数进行交换：\n\nf(x) = (x/50)^(100/(50-x)) = (x/50)^(100/(50-x)) * (50-x)^(100/(50-x)) / (50-x)^(100/(50-x))\n\n再次进行化简：\n\nf(x) = (x * (50-x))^(100/(50-x)) / (50-x)^(100/(50-x))\n\n接下来，对f(x)求导数：\n\nf'(x) = (100/(50-x)) * (x * (50-x))^(100/(50-x) - 1) * (1 * (50-x) - x * (-1)) / (50-x)^(100/(50-x))\n\n化简后得到：\n\nf'(x) = (100/(50-x)) * (x * (50-x))^(100/(50-x) - 1) * (50 - x + x) / (50-x)^(100/(50-x))\n\n化简后得到：\n\nf'(x) = (100/(50-x)) * (x * (50-x))^(100/(50-x) - 1)\n\n由于100/(50-x) - 1是一个常数，所以：\n\nf'(x) = C * (x * (50-x))^(100/(50-x) - 1)\n\n其中C为常数。\n\n由于C为常数，所以f'(x)的符号只取决于(x * (50-x))^(100/(50-x) - 1)的符号。\n\n当x * (50-x) > 0时，(x * (50-x))^(100/(50-x) - 1) > 0，即f'(x) > 0。\n\n当x * (50-x) < 0时，(x * (50-x))^(100/(50-x) - 1) < 0，即f'(x) < 0。\n\n综上所述，当x * (50-x) > 0时，f(x)是单调递增的；当x * (50-x) < 0时，f(x)是单调递减的。