Question

已知0≤x≤2,求函数y=4^(x-1/2)-3*2^x+5的最大值和最小值

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Answer 1

y=2^[2*(x-1/2)]-3*2^x+5
=2^(2x-1)-3*2^x+5
=2^2x/2-3*2^x+5

0<=x<=2
2^0<=2^x<=2^2
令a=2^x
则1<=a<=4

y=(1/2)a^2-3a+5
=(1/2)(a^2-6a)+5
=(1/2)(a^2-6a+9-9)+5
=(1/2)(a^2-6a+9)-9*(1/2)+5
=(1/2)(a-3)^2+1/2
1<=a<=4
所以a=3,y最小=1/2
a=1，y最大=5/2

Answer 2

换元法：令2^x=t(1≤t≤4)
y=1/2t^2-3t+5,对称轴为t=3,开口向上
最大值为f(1)=5/2
最小值为f(3)=1/2

Answer 3

y=2^[2*(x-1/2)]-3*2^x+5
=2^(2x-1)-3*2^x+5
=2^2x/2-3*2^x+5
0<=x<=2
2^0<=2^x<=2^2
令a=2^x
则1<=a<=4
y=(1/2)a^2-3a+5
=(1/2)(a^2-6a)+5
=(1/2)(a^2-6a+9-9)+5
=(1/2)(a^2-6a+9)-9*(1/2)+5
=(1/2)(a-3)^2+1/2
1<=a<=4
所以a=3,y最小=1/2
a=1，y最大=5/2