Question

设函数f(x)=xx-2x+2,x属于[t,t+1]的最小值为g(t),求g(t)的表达式

Answer 1

∵f (x) = x^2-2x+2=（x-1)^2+1
∴对称轴x=1
分类讨论
1.x=1∈[t, t+1]时,即0≤t≤1时,g(t)=1;
2.x=1<t时, 对称轴在区间左侧, f (x)在[t, t+1]上递增,
g(t)=f(t) =t^2-2t+2
3.x=1>t+1即t<0时, 对称轴在区间右侧, f (x)在[t, t+1]递减
则g (t) = f(t+1)=t^2+3

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Answer 2

.x=1<t时, 对称轴在区间左侧, f (x)在[t, t+1]上递增,
g(t)=f(t) =t^2-2t+2
3.x=1>t+1即t<0时, 对称轴在区间右侧, f (x)在[t, t+1]递减
则g (t) = f(t+1)=t^2+3