设a属于R 函数f(x)=ax^3-3x^2 ,x=2是函数y=f(x)的极值点(1)求a2)若函数g(x)=f(x)+f'(x)x属于0到2闭区间x=0处取最大值求a得取值范围

设a属于R 函数f(x)=ax^3-3x^2 ,x=2是函数y=f(x)的极值点(1)求a2)若函数g(x)=f(x)+f'(x)x属于0到2闭区间x=0处取最大值求a得取值范围
设a属于R 函数f(x)=ax^3-3x^2 ,x=2是函数y=f(x)的极值点(1)求a
2)若函数g(x)=f(x)+f'(x)x属于0到2闭区间x=0处取最大值求a得取值范围

设a属于R 函数f(x)=ax^3-3x^2 ,x=2是函数y=f(x)的极值点(1)求a2)若函数g(x)=f(x)+f'(x)x属于0到2闭区间x=0处取最大值求a得取值范围
(1) f'(x) = 3ax²-6x = 3x(ax - 2) = 0,显然x = 2为ax - 2 = 0的解,2a - 2 = 0,a = 1
(2) g(x) = f(x) + f'(x) = ax³ - 3x² + 3ax²-6x = ax³ + 3(a - 1)x²- 6x
g'(x) = 3ax² + 6(a -1)x - 6
在[0,2]内,x = 0处g(x)取最大值,则g'(x)在[0,2]内0,a < 0,x = (1 - a)/a < 0
对称轴在y轴左侧
在[0,2]内,g(x)在x=0处取最大值,须下列2个条件同时成立:
(a) g'(0)≤0,g'(0) = -6 < 0总成立
(b) g'(2)≤0,g'(2) = 6(4a - 3) ≤0,a ≤ 3/4
结合前提a < 0,得a < 0
(iii) a = 0
g(x) = -3x² - 6x = -3x(x + 2)
抛物线开口向下,对称轴x = -1
g(0) = 0,在[0,2]内,g(x)在x=0处取最大值
结合(i)(ii)(iii):a ≤ 3/4