(1)
1 ≤ x ≤ 3时, 线段斜率 = (35 - 25)/(3 - 1) = 5, 线段方程y-25 = 5(x - 1), y = 5x + 20
3 < x ≤ 8时, y = 35
8 < x ≤ 10时, 线段斜率 = (25 - 35)/(10 - 8) = -5, 线段方程y - 35 = -5(x - 8), y = 75 - 5x
s, x的关系显然是抛物线,顶点为(5, 16.5), s = a(x - 5)² + 16.5
抛物线过(10, 4): 4 = a(10 - 5)² + 16.5
a = -0.5
s = -0.5(x - 5)² + 16.5 = -0.5x² + 5x + 4
(2)
每件利润W = 每件售价(y) - 每件成本(s)
(i) 1 ≤ x ≤ 3时:
W = 5x + 20 - (-0.5x² + 5x + 4) = 0.5x² + 16
此为对称轴为x = 0, 开口向上的抛物线的右支, W随着x的增大而增大
(ii) 3 < x ≤ 8时
W = 35 - (-0.5x² + 5x + 4) = 0.5x² - 5x + 31 = 0.5(x - 5)² + 18.5
此为对称轴为x = 5, 开口向上的抛物线
3 < x ≤ 5时, 此为抛物线的左支, W随着x的增大而减小
5 < x ≤ 8时, 此为抛物线的右支, W随着x的增大而增大
(iii)8 < x ≤10时
W = 75 - 5x - (-0.5x² + 5x + 4) = 0.5x² - 10x + 71
= 0.5(x - 20)² + 21
此为对称轴为x = 20, 开口向上的抛物线的左支, W随着x的增大而减小
1 ≤ x < 3或5 ≤ x < 8时, W随着x的增大而增大
(3) 显然只须比较第3个月和第8个月即可
x = 3, W = 0.5*3² + 16 = 20.5
x = 8, W = 0.5(8 - 5)² + 18.5 = 23
第8个月每件利润最大(23元）