Question

对方程x^3+xy^2-4y^2=0,当y>=0时,函数y=f(x)的单调性（请用求导的方法解答,

Answer 1

x³ + xy² - 4y² = 0

y² = x³/(4 - x) > 0, 定义域: 0 < x < 4

两边对x求导: 2yy' = 3x²(4 - x)⁻¹ + x³(-1)(4 - x)⁻²(-1) = x²[3(4 - x) + x](4 - x)⁻²

= 2x²(6 - x)/(4 - x)²

y' = x²(6 - x)/[y(4 - x)²]

x² > 0, (4 - x)² > 0, y > 0

y' = 0, x = 6, 在定义域外

0 < x < 4: 6 - x > 0, y' > 0

在定义域内单调递增