1.f(1+x) = a(1+x)² + b(1+x) + 2 = ax² + (2a+b)x + a²+b+2
f(1-x) = a(1-x)² + b(1-x) + 2 = ax² -(2a+b)x + a²+b+2
2a+b = -(2a+b)
2a+b = 0,b = -2a
f(3) = 9a+3b+2 = 5
3a+b = 1
3a-2a = 1
a = 1
b = -2
f(x) = x² -2x + 2
(2)题目不是很清楚,假定为g(x) = f(x)e^x -(1/3)ex³
g'(x) = f'(x)e^x + f(x)(e^x)' - ex^2
= (2x-2)e^x + (x²-2x+2)e^x - ex^2
= x²(e^x - e) = 0
e^x =e,即x = 1时,g(x)取极值:g(1) = f(1)e - (1/3)e = e - e/3 = 2e/3