问题 解答题

设函数f(x)=a2x2(a>0).

(1)将函数y=f(x)图象向右平移一个单位即可得到函数y=φ(x)的图象,写出y=φ(x)的解析式及值域;

(2)关于x的不等式(x-1)2>f(x)的解集中的整数恰有3个,求实数a的取值范围.

答案

(1)∵函数f(x)=a2x2(a>0),将函数y=f(x)图象向右平移一个单位可得到函数y=φ(x)的图象,

∴y=φ(x)的解析式为:y=φ(x)=a2(x-1)2,由完全平方非负的特点可知其值域为:[0,+∞)

(2)解法一:不等式(x-1)2>f(x)的解集中的整数恰有3个⇔(1-a2)x2-2x+1>0恰有三个整数解,

故1-a2<0.令h(x)=(1-a2)x2-2x+1,由h(0)=1>0且h(1)=-a2<0(a>0)

所以函数h(x)=(1-a2)x2-2x+1的一个零点在区间(0,1),另一个零点一定在区间[-3,-2)

h(-2)>0
h(-3)≤0
解得
4
3
≤a≤
3
2

解法二:(1-a2)x2-2x+1>0恰有三个整数解,故1-a2<0,即a>1

(1-a2)x2-2x+1=[(1-a)x-1][(1+a)-1]>0

所以

1
1-a
<x<
1
1+a
,又因为0<
1
1+a
<1

所以-3≤

1
1-a
<-2,解得
4
3
≤a≤
3
2

单项选择题
单项选择题

Even plants can run a fever, especially when they are under attack by insects or disease. But (1) humans, plants can have their temperature (2) from 3,000 feet away—straight up. A decade ago, (3) the infrared scanning technology developed for military purpose and other satellites, physicist Stephen Paley (4) a quick way to take the temperature of crops to determine (5) ones are under stress. The goal was to let farmer (6) target pesticide spraying (7) rain poison on a whole field, which (8) include plants that don’t have the pest problem.

Even better, Paley’s Remote Scanning Services Company could detect crop problem before they became (9) to the eye. Mounted on a plane flown at 3,000 feet (10) , an infrared scanner measured the heat emitted by crops. The data were (11) into a color-coded map showing (12) plants were running "fevers". Farmers could then spot spray, using 50 to 70 percent less pesticide than they (13) would.

The bad news is that Paley’s company closed down in 1984, after only three years. Farmers (14) the new technology and long-term backers were hard (15) . But with the renewed concern about pesticides on produce, and refinements in infrared scanning, Paley hopes to (16) into operation. Agriculture experts have no doubt about the technology works. "This technique can be used (17) 75 percent of agricultural land in the United States," says George Oerther of Texas A&M. Ray Jackson, who recently retired from the Department of Agriculture, thinks (18) infrared crop scanning could be adopted by the end of the decade. But (19) Paley finds the financial backing (20) he failed to obtain 10 years ago.

13()

A.otherwise

B.still

C.thus

D.therefore