已知变量x,y,z满足不等式组\left\{ \begin{array}{l}{x+2y-1≥0} \\ {2x+y-2≤0} \\ {x-y+2≥0},则z={8}^{x}+{2}^{y}的最小值

已知变量x,y,z满足不等式组\left\{ \begin{array}{l}{x+2y-1≥0} \\ {2x+y-2≤0} \\ {x-y+2≥0},则z={8}^{x}+{2}^{y}的最小值
已知变量x,y,z满足不等式组\left\{ \begin{array}{l}{x+2y-1≥0} \\ {2x+y-2≤0} \\ {x-y+2≥0},则z={8}^{x}+{2}^{y}的最小值

已知变量x,y,z满足不等式组\left\{ \begin{array}{l}{x+2y-1≥0} \\ {2x+y-2≤0} \\ {x-y+2≥0},则z={8}^{x}+{2}^{y}的最小值
设a = log_2(6),则2^a = 6.
8^x+2^y = 2^(3x)+2^a·2^(y-a)
= 2^(3x)+6·2^(y-a)
= 2^(3x)+2^(y-a)+2^(y-a)+2^(y-a)+2^(y-a)+2^(y-a)+2^(y-a)
≥ 7·(2^(3x)·2^(y-a)·2^(y-a)·...·2^(y-a))^(1/7)
= 7·2^((3x+6y-6a)/7)
= 7·2^((3(x+2y-1)+3-6a)/7)
≥ 7·2^((3-6a)/7) (x+2y-1 ≥ 0)
= 7·(2^(1-2a))^(3/7)
= 7/18^(3/7).
当3x = y-a,x+2y-1 = 0时等号成立.
解得x = (1-2a)/7,y = (3+a)/7.
此时,可验证2x+y-2 = -3(3+a)/7 ≤ 0,x-y+2 = 3(4-a)/7 ≥ 0.
即x = (1-2a)/7,y = (3+a)/7满足不等式组.
于是z = 8^x+2^y的最小值就是7/18^(3/7).