数学已知定义在正实数集上的函数f
(1)
f(x)=1/2x^2+2x,g(x)=3lnx+b
f'(x)=x+2,g'(x)=3/x
设公共点(m,n)
f'(m)=g'(m),m+2=3/m,m=1
n=f(1)=1/2+2=5/2
g(1)=b=f(1)=5/2
b=5/2
(2)
f'(x)=x+2a,g'(x)=3a^2/x
设公共点(m,n)
f'(m)=g'(m),m+2a=3a^2/m
m^2+2am-3a^2=0,(m+3a)(m-a)=0
m=a
f(m)=g(m),f(a)=g(a)
1/2a^2+2a^2=3a^2lna+b
b=5/2a^2-3a^2lna
设h(a)=5/2a^2-3a^2lna...全部
(1)
f(x)=1/2x^2+2x,g(x)=3lnx+b
f'(x)=x+2,g'(x)=3/x
设公共点(m,n)
f'(m)=g'(m),m+2=3/m,m=1
n=f(1)=1/2+2=5/2
g(1)=b=f(1)=5/2
b=5/2
(2)
f'(x)=x+2a,g'(x)=3a^2/x
设公共点(m,n)
f'(m)=g'(m),m+2a=3a^2/m
m^2+2am-3a^2=0,(m+3a)(m-a)=0
m=a
f(m)=g(m),f(a)=g(a)
1/2a^2+2a^2=3a^2lna+b
b=5/2a^2-3a^2lna
设h(a)=5/2a^2-3a^2lna
h'(a)=5a-3a-6alna=2a(1-3lna)
当1-3lna>0,00,h(x)单调增
当1-3lnae^(1/3),h'(x)<0,h(x)单调减
当1-3lna=0,a=e^(1/3),h'(x)=0,h(x)取极大值
b最大值=h(e^(1/3))
=5/2*e^(2/3)-e^(2/3)=3/2*e^(2/3)
。
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