Question

已知f（x)=ax平方+bx，满足1小于等于f（-1）小于等于2且2小于等于f（1）小于等于4，求f（-2）的取值范围。

Answer 1

1小于等于f（-1）小于等于2
则1≤a-b≤2 (1)
2小于等于f（1）小于等于4
则2≤a+b≤4 (2)
f(-2)=4a-2b (3)
设f(-2)=m(a-b)+n(a+b)
=(m+n)a+(-m+n)b (4)
则(3)(4)对比m+n=4 -m+n=-2
解得m=3 n=1
所以f(-2)=3(a-b)+(a+b)≥3*1+2=5
f(-2)=3(a-b)+(a+b)≤3*2+4=10
故f（-2）的取值范围为[5,10]
希望能帮到你，祝学习进步O(∩_∩)O

Answer 2

1 <= f(-1) <= 2 <= f(1) <= 4
1 <= a-b <= 2 <= a+b <= 4
3/2 <= a <= 3 ==> 6 <= 4a <= 12
0 <= b <= 1 ==> -2 <= -2b <= 0
f(-2) = 4a - 2b
4 <= f(-2) <= 12