Question

二次函数y=ax^2+bx+c中,a:b:c=2:3:4，且这个函数的最小值为23/4，则这个二次函数为

Answer 1

二次函数y=ax^2+bx+c中,a:b:c=2:3:4，且这个函数的最小值为23/4，则有
a>0,
c-b²/(4a)=23/4
∵a:b:c=2:3:4
∴b=3a/2,c=4b/3
代入c-b²/(4a)=23/4解得b=6
∴a=4,c=8
这个二次函数为y=4x²+6x+8

Answer 2

根据最小值为 23/4，我们知道，抛物线开口向上，a > 0，
二次函数 y = a(x - h)" + k 是 k = 23/4
我们再把一般式 y = ax" + bx + c 做配方看看
y = a[ x" + (b/a)x ] + c
= a[ x" + 2(b/2a)x + (b/2a)" - b"/4a" ] + c
= a( x + b/2a )" - b"/4a + 4ac/4a
= a( x + b/2a )" + (4ac - b")/4a
那么，(4ac - b")/4a = 23/4

根据 a:b:c = 2:3:4，设 a= 2d，b= 3d，c= 4d，则
[ 4(2d)(4d) - (3d)" ] / 4(2d) = 23/4
[ 32d" - 9d" ] / 8d = 23/4
23d" / 8d = 23/4
23d / 8 = 23/4
d = 2
a = 2d = 4
b = 3d = 6
c = 4d = 8
y = 4x" + 6x + 8

检验看看
y = 4x" + 6x + 8
= 4[ x" + 3/2x + (3/4)" - 9/16 ] + 8
= 4[ x + 3/4 ]" - 9/4 + 32/4
= 4( x + 3/4 )" + 23/4
没错