The minimum value of the function f (x) = x2-4x-4 in the closed interval [T, t + 1] (t ∈ R) is denoted as G (T). (1) try to write the functional expression of G (T). (2) make the image of G (T) and find the minimum value of G (T)

The minimum value of the function f (x) = x2-4x-4 in the closed interval [T, t + 1] (t ∈ R) is denoted as G (T). (1) try to write the functional expression of G (T). (2) make the image of G (T) and find the minimum value of G (T)

(1) Since the axis of symmetry of the function f (x) = x2-4x-4 is x = 2, when 2 < T, the function f (x) increases monotonically in the closed interval [T, t + 1], so the minimum value of the function g (T) = ft) = t2-4t-4 When t + 1 ＜ 2, i.e. t ＜ 1, the function f (x) decreases monotonically in the closed interval [T, t + 1], so the minimum value of the function g (T) = ft + 1) = t2-2t-7. To sum up, G (T) = T2 − 4T − 4 & nbsp;, & nbsp; t > 2 − 8 & nbsp;, & nbsp;, 1 ≤ t ≤ 2t2 − 2T − 7 & nbsp; & nbsp; (2) make an image of G (T), as shown in the figure: the minimum value of G (T) is - 8