Question

函数y=[log1/4(x)]^2-log1/4(x)^2+5,x∈[2,4]的值域是_____

其中1/4为底数，x为真数
网上有答案，但是看不懂，希望数学高手帮忙解答，如果答的好，有5~10加分哟~~

Answer 1

y=[log(1/4)x]^2-log(1/4)x^2+5
=[log(1/4)x]^2-2log(1/4)x+5
=[log(1/4)x-1]^2+4
∵x∈[2,4],log(1/4)x-1<0
∴在区间[2,4]单调递减
当x=2时,
y=[log(1/4)2-1]^2+4=13
当x=4时,
y=[log(1/4)4-1]^2+4=8
y∈[8,13]

追问

好像答案不对吧

追答

是错了，抱歉
y=[log(1/4)x]^2-log(1/4)x^2+5
=[log(1/4)x]^2-2log(1/4)x+5
=[log(1/4)x-1]^2+4
∵x∈[2,4],log(1/4)x<0
∴在区间[2,4]单调递增
当x=2时,
y=[log(1/4)2-1]^2+4=25/4
当x=4时,
y=[log(1/4)4-1]^2+4=8
y∈[25/4,8]