Let f : R → R be given by f(x) = { x⁵ + 5x⁴ + 10x³ + 10x² + 3x - 1, x < 0
x² - x + 1 0 ≤ x < 1
(2/3)x³ - 4x² + 7x - (8/3) 1 ≤ x < 3
(x - 2)ln(x - 2) - x + (10/3) x ≥ 3. Then which of the following options is/are Correct?

Correct option is D. f′ has a local maximum at x=1
f(x) = x⁵ + 5x⁴ + 10x³ + 10x² + 3x - 1; x < 0
x² - x + 1 0 ≤ x < 1
(2/3)x³ - 4x² + 7x - 8/3 1 ≤ x < 3
(x - 2)ln(x - 2) - x + 10/3 x ≥ 3
f′(x) = 5(x + 1)⁴ - 2; x < 0
2x - 1; 0 ≤ x < 1
2x² - 8x + 7 1 ≤ x < 3
ln(x - 2) x ≥ 3
x⁵ + 5x⁴ + 10x³ + 10x² + 3x - 1 takes value between −∞ to 1
Also (x - 2)ln(x - 2) - x + 10/3 takes value between 1/3 to ∞
So, range of f(x) is R. So option (A) is correct
f′′(1−) = 2 and f′′(1+) = −4 so f′(x) is non-diff at x = 1 so option (B) is correct
f′(x) has local maxima at x = 1 so option (C) is correct.