Find the inflection point of the cubic curve f(x) = 2x^3 - 6x^2 + 5x - 1.

Answer and explanation

Correct answer: C. (1, 0)

  1. Step 1: Compute the first derivative: f'(x) = 6x^2 - 12x + 5.
  2. Step 2: Compute the second derivative: f''(x) = 12x - 12.
  3. Step 3: Set f''(x) = 0: 12x - 12 = 0, giving x = 1.
  4. Step 4: Evaluate y at x = 1: f(1) = 2(1)^3 - 6(1)^2 + 5(1) - 1 = 2 - 6 + 5 - 1 = 0.
  5. Step 5: Since f''(x) changes sign around x = 1, the inflection point is (1, 0).
Answer
(1, 0).
Why not the others: '(2, 1)' evaluates an incorrect x-coordinate; '(0, -1)' is the y-intercept, not where concavity changes; '(1, 2)' miscalculates the y-value as 2 - 6 + 5 + 1.

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