Question

已知函数f(x)=xlnx,g(x)=x^3 ax^2-x 2,若不等式2f(x)小于等于g(t) 2恒成立，求实数a的取值范围

已知函数f(x)=xlnx,g(x)=x^3+ax^2-x+2,若不等式2f(x)小于等于g(t)+2恒成立，求实数a的取值范围

Answer 1

f(x)=xlnx
g(x)=x^3+2ax^2+2
x>0,2f(x)<=g(x)+2
x>0，g(x)+2-2f(x)>=0
F(x)=g(x)+2-2f(x)=x^3+2ax^2+4-2xlnx
F'(x)=3x^2+4ax-2lnx-2
F''(x)=6x+4a-2/x,
x=根号(a^2-3)-a
x>0
2f(x)≤g(x)+2
a≤-1.3256

Answer 2

2f(x)≤g'(x)+2
==>g'(x)+2-2f(x)≥0
==>3x^2+2ax-1+2-2xlnx≥0
==>3x+2a+1/x-2lnx≥0
==>3x+1/x-2lnx≥-2a
==>lnx-3x/2-1/2x≤a
令h（x)=lnx-3x/2-1/2x==>h(x)≤a
h'(x)=1/x-3/2+1/2x^2=0
==>3x^2-2x-1=0
==>(3x+1)(x-1)=0 x∈（0，+∞）
==>x=1
当x=1时h(x)有最小值（-2）
∴a≥-2