Question

已知函数f(x)=3^x,且f(a)=2,g(x)=3^ax-4^x

(1)求g(x)的解析式
（2）当x属于[-2,1]时，求g（x)的值域

Answer 1

(1)f(a)=2,3^a=2,a=log3(2)

g(x)=3^(xlog3(2))-4^x=3^(log3(2^x))-4^x=2^x-4^x

(2)令2^x=t,x属于[-2,1]所以t属于[1/4,2]
g(x)=t-t^2,则g(x)属于[-2,1/4]

Answer 2

1.
f(a) = 3^a = 2
a = log(3)2
g(x)=3^ax-4^x
= 3^(xlog(3)2) - 4^x
= 2^x - 4^x

2.
g'(x) = (ln2) 2^x + (ln4) 4^x
= ln2( 2^x - 2(4^x))
= (ln2)2^x( 1- 2^(x+1))

g'(x) =0, => x=-1
g''(x) = ln2( (ln2)2^x - 2ln4(4^x))
= 2^x . (ln2)^2(1-2^(x+2))

g''(-1)<0 ( max)
g(-1) = 2^(-1) - 4^(-1)
= 1/4

g(-2) = 2^(-2) - 4^(-2)
= 1/4 - 1/16
= 3/16

g(1) = 2^(1) - 4^(1)
= -2

max g(x) = g(-1) = 1/4
min g(x) = g(1) = -2

range= [-2, 1/4]

Answer 3

f(x)=3^x，所以f(a)=3^a
已知f(a)=2，所以a^3=2
将a^3=2代入g(x)=3^ax-4^x=2x-4^x
(1)g(x)=2x-4^x
(2)当x属于[-2,1]时，求g（x)的值域
g（x)的的一阶导数=2-4^xln4
由2-4^xln4=0求出驻点x=0
将x=0代入g(x)=2x-4^x=2*0-4^0=-1
将x=-2代入g(x)=2x-4^x=2*（-2）-4^(-2)=-65/16
将x=1代入g(x)=2x-4^x=2*1-4^1=-2
综上计算结果g（x)的值域[-65/16,-1]

Answer 4

由f(x)=3^x,f(a)=2可得
3^a=2
所以g(x)=3^ax-4^x=2^x-4^x=-(2^x-1/2)^2+1/4
当x属于[-2,1]时
2^x属于[1/4,1]
所以g(x)的值域为[0,1/4]

Answer 5

抱歉，看错题了~