数学高手请进!已知定义在正实数上的函数f(x)=1/2x^2+2ax,g(x)=3a^2lnx+b,其中a>.设曲线y=f（x）,g=（x）由公共点,且在该点处的切线相同.（1）用a表示b,并求b的最大值（2）求证：f(x)>=g(x)(x>0)

Question

数学高手请进!
已知定义在正实数上的函数f(x)=1/2x^2+2ax,g(x)=3a^2lnx+b,其中a>.设曲线y=f（x）,g=（x）由公共点,且在该点处的切线相同.
（1）用a表示b,并求b的最大值
（2）求证：f(x)>=g(x)(x>0)

Answer 1

（1）两函数切线相同,说明函数导数相同,函数值相同：
f'(x)=g'(x)
x + 2a = 3 a^2/x
x^2 + 2ax - 3a^2 = 0
(x - a)(x + 3a) = 0
x=a
x=-3a (因a >0,x < 0 舍去)
f(a) = g(a)
1/2 a^2 + 2a^2 = 3a^2 lna + b
b = 5a^2/2 - 3a^2lna
b'= 5a - (6a lna + 3a^2/a)
b'= 5a - (6a lna + 3a)
b'= 2a - 6a lna
b'=0
2a - 6a lna = 0
lna = 1/3
a = e^(1/3)
Max b = 5/2 * e^(2/3) - e^(2/3)
Max b = 3/2 e^(2/3)
(2)
h(x)=f(x)-g(x)
=1/2x^2+2ax - (3a^2lnx +b)
=1/2x^2+2ax - (3a^2lnx + 5a^2/2 - 3a^2lna)
=1/2x^2+2ax - 5a^2/2 + 3a^ln(a/x)
=1/2(x^2 + 4ax - 5a^2) + 3a^2(a/x)
=1/2(x-a)(x + 5a) + 3a^2(a/x)
当：x=a时,h(a)= 0
h'(x) = x + 2 a - 3a^2/x = (x^2 + 2ax -3a^2)/x = (x-a)(x+3a)/x
x>a>0 h(x)>0
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