Question

已知函数f（x）=2x^3+ax与g(x)=bx^2+c

已知函数f（x）=2x^3+ax与g(x)=bx^2+c的图象都经过p（2，0）且在点P处上有相同的切线
（1）求实数a、b、c
（2）设函数F（x）=f（x）+g（x） 求F（x）
在该区间上的单调性

Answer 1

f(x) = 2x^3 + ax,
f'(x) = 6x^2 + a

g(x) = bx^2 + c,
g'(x) = 2bx

16 + 2a = 0,
4b + c = 0,
24 + a = 4b.

a = -8,
b = 6 + a/4 = 4,
c = -4b = -16.

F(x) = 2x^3 + ax + bx^2 + c

= 2x^3 + 4x^2 - 8x - 16,

F'(x) = 6x^2 + 8x - 8 = 2[3x^2 + 4x - 4] = 2(3x - 2)(x + 2)

x < -2 时，F'(x)>0. F(x)单调递增
-2 < x < 2/3时，F'(x) < 0. F(x)单调递减
x > 2/3时，F'(x)>0. F(x)单调递增