How many inflection points does the function f(x) = x^4 - 4x^3 + 6x^2 have?
Answer and explanation
Correct answer: D. 0
- Step 1: First derivative: f'(x) = 4x^3 - 12x^2 + 12*x.
- Step 2: Second derivative: f''(x) = 12x^2 - 24x + 12 = 12(x^2 - 2x + 1) = 12*(x - 1)^2.
- Step 3: An inflection point requires that f''(x) changes sign.
- Step 4: Since 12*(x - 1)^2 >= 0 for all x, f''(x) does not change sign around x = 1. Therefore, there are 0 inflection points.
Answer
0.
Why not the others: 1 misidentifies x = 1 as an inflection point without checking sign change; 2 assumes every degree 4 polynomial has 2 inflection points; 3 confuses roots of f' with inflection points.
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