1995 AMC 12 第 12 题

Question

设

f

是线性函数，满足

f(1)\le f(2)

、

f(3)\ge f(4)

，且

f(5)=5

。下列哪个陈述正确？

Let

f

be a linear function with the properties that

f(1)\le f(2),

f(3)\ge f(4),

and

f(5)=5.

Which of the following statements is true?

f(0)\lt0

f(0)=0

f(1)\lt f(0)\lt f(-1)

f(0)=5

f(0)\gt5

将每个不等式转化为对斜率的限制

Translate each inequality into a condition on the slope

两个斜率条件合起来可以确定整个线性函数

The two slope conditions together determine the entire linear function

由

f(1)\le f(2)

可知斜率非负；由

f(3)\ge f(4)

可知斜率非正。因此斜率为

0

，所以

f

是常函数。由于

f(5)=5

，有

f(0)=5

。因此正确答案是 D。

From

f(1)\le f(2),

the slope is nonnegative. From

f(3)\ge f(4),

it is nonpositive. Thus the slope is

0,

so

f

is constant. Since

f(5)=5,

we have

f(0)=5.

Thus the correct answer is D.

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