已知函数f(x)=x^2+ax+Inx,其中a∈R
1个回答
f(x)=x^2+ax+Inx
x>0
1
a=-3
f(x)=x^2-3x+Inx
f'(x)=2x-3+1/x
令f'(x)=0
2x-3+1/x=0
2x^2-3x+1=0
(x-1)(2x-1)=0
x=1 x=1/2
f''(x)=2-1/x^2
f''(1)=2-1=1>0 f(1)为极小值
f''(1/2)=2-4=-20 9a^2>8 a^2>9/8
x1=(3a+(9a^2-8)^0.5)/4
x2=(3a-(9a^2-8)^0.5)/4
M=f(x1)+f(x2)
=(3a+(9a^2-8)^0.5)^2/16+a(3a+(9a^2-8)^0.5)/4+In(3a+(9a^2-8)^0.5)/4+(3a-(9a^2-8)^0.5)^2/16+a(3a-(9a^2-8)^0.5)/4+In(3a-(9a^2-8)^0.5)/4
=(18a^2+18a^2-16)/16+6a^2/4+ln(9a^-9a^+8)/16
=9a^2/4-1+6a^2/4+ln1/2
=15a^2/4-1+ln1/2>103/32+ln1/2
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