(1)
f(x)=x²+4ax+2a+6=(x+2a)²-4a²+2a+6
顶点坐标(-2a,-4a²+2a+6),二次项系数1>0,函数图象开口向上.
当x=-2a时,f(x)有最小值f(x)min=-4a²+2a+6,又函数值域为[0,+∞)
-4a²+2a+6=0
整理,得
2a²-a-3=0
(a+1)(2a-3)=0
a=-1或a=3/2
(2)
f(x)≥0恒成立,方程x²+4ax+2a+6=0判别式△≤0
△=(4a)²-4(2a+6)≤0
整理,得
2a²-a-3≤0
(a+1)(2a-3)≤0
-1≤a≤3/2
a+3>0
f(a)=2-a|a+3|=2-a(a+3)=-a²-3a+2=-(a+3/2)²+17/4
-1≤a≤3/2,函数单调递减
当a=-1时,f(a)有最大值f(a)max=-1+3+2=4
当a=3/2时,f(a)有最小值f(a)min=-(3/2)²-3(3/2)+2=-19/4
函数f(a)的值域为[-19/4,4]