1.求函数y=ln(x^2+3x+2)的n阶导数dny/dx^n.明天上午必须得出答案!2.讨论函数f(x)=x^4-2x^3+1的凹凸性与拐点.

Question

1.求函数 y=ln(x^2+3x+2)的n阶导数dny/dx^n.明天上午必须得出答案!
2.讨论函数f(x)=x^4-2x^3 +1的凹凸性与拐点.

Answer 1

y = ln(x²+3x+2)
dy/dx = (2x+3)/(x²+3x+2) = (2x+3)/[(x+2)(x+1)] = (x+2)^(-1) + (x+1)^(-1),n = 0
d²y/dx² = (-1)(1!)(x+2)^(-2) + (-1)(1!)(x+1)^(-2),n = 1
d³y/dx³ = (-1)²(2!)(x+2)^(-3) + (-1)²(2!)(x+1)^(-3),n = 2
d^ny/dx^n = [(-1)^(n-1)][(n-1)!](x+2)^(-n) + [(-1)^(n-1)][(n-1)!](x+1)^(-n),n = n-1
d^(n+1)/dx^(n+1) = [(-1)^n](n!)(x+2)^[-(n+1)] + [(-1)^n](n!)(x+1)^[-(n+1)],n = n
∴d^ny/dx^n = [(-1)^(n-1)][(n-1)!](x+2)^(-n) + [(-1)^(n-1)][(n-1)!](x+1)^(-n)
f(x) = x⁴ - 2x³ + 1
f'(x) = 4x³ - 6x²
f''(x) = 12x² - 12x = 12x(x - 1)
Let f''(x) = 0
=> x(x - 1) = 0 => x = 0 or x = 1
x