Question

已知x^2+x小于等于6求y=1/4^x-1/2^x+1的最大值和最小值并求出对应的x

Answer 1

x^2+x≤6
x^2+x-6≤0
(x+3)(x-2)≤0
-3≤x≤2

y = 1/4^x-1/2^x+1
= (1/2)^(2x) - (1/2)^x + 1
= [ (1/2)^x - (1/2) ] ^2 - 1/4 + 1
= [ (1/2)^x - (1/2) ] ^2 + 3/4

当(1/2)^x=1/2，即x=1时有最小值：ymin = f(1)=0+3/4=3/4
当x＜1时单调减，当x＞1时单调增
f(-3) = 1/4^(-3)-1/2^(-3)+1 = 64-8+1 = 57
f(2) = 1/4^2-1/2^2+1 = 1/16-1/4+1 = 13/16
x=-3时有最大值ymax=f(-3)=57

Answer 2

x^2+x<=6
x^2+x-6<=0
-3=<x<=2
y=1/4^x-1/2^x+1
令1/2^x=t,t∈[1/4,8]
y=t^2-t+1=(t-1/2)^2+3/4,t=1/2,x=1时y=1/4^x-1/2^x+1取得最小值3/4
当t=8,x=-3时,y=t^2-t+1=64-8+1=57,是取大值.

Answer 3

x^2+x-6≤ 0
(x+3)(x-2)≤ 0
-3≤ x≤ 2
当x=-3,y=1/4^x-1/2^x+1的最大值 64-8+1=57
当x=2,y=1/4^x-1/2^x+1的最小值 1/16-1/4+1=13/16