Question

已知0≤x≤2,则y=4(x-1/2)-3·2^x+5的最大值为

求下图中阴影部分面积

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