Given the function f (x) = x2 + 2aX + 2, X ∈ [- 5,5] (1) when a = - 1, find the maximum value of the function; (2) find the value range of the real number a, so that y = f (x) is a monotone function in the interval [- 5,5]; (3) find the minimum value of y = f (x)

Given the function f (x) = x2 + 2aX + 2, X ∈ [- 5,5] (1) when a = - 1, find the maximum value of the function; (2) find the value range of the real number a, so that y = f (x) is a monotone function in the interval [- 5,5]; (3) find the minimum value of y = f (x)

(1) When a = - 1, f (x) = x2-2x + 2 = (x-1) 2 + 1, ｜ when x = 1, f (x) min = f (1) = 1; when x = - 5, f (x) max = 37; (2) ∵ the image of F (x) = x2 + 2aX + 2 is parabolic, and the opening is upward, and the axis of symmetry is x = - A; (3) ∵ the image of F (x) = x2 + 2aX + 2 is parabolic, and the opening is upward When a ≥ 5, f (x) is an increasing function on [- 5,5]; when a ＞ a ＞ - 5, f (x) min = f (- 5) = 27-10a; when 5 ＞ a ＞ - 5, f (x) is a decreasing function on [- 5,5]; when a ＜ - 5, f (x) min = f (- a) = - A2 + 2; when a ≤ - 5, f (x) is a decreasing function on [- 5,5]; when f (x) min = f (5) = 27 + 10A; therefore, the minimum value of f (x) on [- 5,5] is: F ）min=27−10a(a≥5)−a2+2(5＞a＞−5)27+10a(a≤−5)．