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For f(x) = (x - 1)² (x - 2) + 1 defined on [0, 2], which statement is true?

1. (A) Range of f is [23/27, 1]
2. (B) The turning points of y = f(x) occur at (1, 1) and (5/3, 23/27).
3. (C) The values of p for which f(x) = p has 3 distinct solutions lie in (23/27, 1).
4. (D) The area enclosed by y = f(x), the lines x = 0 and y = 1 as x varies from 0 to 1 is 7/12.

Correct answer: (B) The turning points of y = f(x) occur at (1, 1) and (5/3, 23/27).

Solution

f'(x) = (x-1)(3x-5) = 0 gives x = 1 and x = 5/3, with f(1) = 1 and f(5/3) = 23/27, so the turning points (1,1) and (5/3,23/27) are correct; the range over [0,2] is [0,1] (f(0)=0), which makes (A) false.

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