If f(x) = x³ - 6x² + ax + b is such that f(2) = 0 and f'(2) = 0, find the values of a and b.
Correct: C
Given f(x) = x³ - 6x² + ax + b. Then f'(x) = 3x² - 12x + a. Since f(2) = 0, we have 2³ - 6(2²) + 2a + b = 0, which simplifies to 8 - 24 + 2a + b = 0, or 2a + b = 16. Since f'(2) = 0, we have 3(2²) - 12(2) + a = 0, which simplifies to 12 - 24 + a = 0, so a = 12. Substituting a = 12 into 2a + b = 16, we get 2(12) + b = 16, so 24 + b = 16, which means b = -8. a=12, b = -8, This should be a = 12, b=-8.