f(x)=ax^3+bx^2+cx+d在x=-2和x=2/3处取极值,解不等式f(-3-2x^2)>f(-x^2+2x-4)

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f(x)=ax^3+bx^2+cx+d在x=-2和x=2/3处取极值,解不等式f(-3-2x^2)>f(-x^2+2x-4)f(x)=ax^3+bx^2+cx+d在x=-2和x=2/3处取极值,解不等

f(x)=ax^3+bx^2+cx+d在x=-2和x=2/3处取极值,解不等式f(-3-2x^2)>f(-x^2+2x-4)
f(x)=ax^3+bx^2+cx+d在x=-2和x=2/3处取极值,解不等式
f(-3-2x^2)>f(-x^2+2x-4)

f(x)=ax^3+bx^2+cx+d在x=-2和x=2/3处取极值,解不等式f(-3-2x^2)>f(-x^2+2x-4)
由于有两个极值点,从而a不为零.求导得
f'(x)=3ax^2+3bx+c,
由条件知,x=-2和x=2/3是f'(x)=0的两个根.
于是
(1)若a>0,令f‘(x)>0,解得 x2/3,
即f(x)在(-无穷,-2)和(2/3,+无穷)上是增函数,
由于 -3-2x^2,-x^2+2x-4都属于(-无穷,-2),
从而原不等式可化为
-3-2x^2>-x^2+2x-4
即x^2+2x-1